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abixiBon^   ^Ipxtn   €ouxni^, 


THE 


COMPLETE 


A  L  G  E  B  E  A, 


DESIGNED  FOR  USB  IN 


SCHOOLS,  ACADEMIES,  AND  COLLEGES. 


BY 

JOSEPH    FICKLIN,    Ph.D., 

FB07B8S0B  OT  ICATHSUATICS  IN  THE  UNIVBR8ITT  OP  THE  STATE  OF  MISSOUHL 


IVISON,  BLAKEMAN,  TAYLOE  &  CO., 
NEW  YORK  AND  CHICAGO. 


ROBINSON'S 

Shorter  Course. 


FIRST  BOOK  IN  ARITHME  TIC.  Primary. 
COMPLETE  ARITHMETIC.  In  One  volume* 
COMPLETE  ALGEBRA. 

ARITHMETICAL  PROBLEMS.    Oral  and  Written. 
ALGEBRAIC  PROBLEMS. 

KEYS  to  Complete  Arithmetic  and  Problems,  and 

to  Complete  Algebra  end  Problems^ 

in  separate  volumes^  for  Teachers, 


AHtlnnetic,  orai:  and  written,  usually  taught  iiv 
THREE  hooks,  is  now  offered,  complete  and  thorough, 
in  ONE  hooh,  "  the  complete  arithmetic' 

*  This  Complete  Arithmetic  is  also  published  in  two  volumes.  TART  J. 
and  PART  11,  are  each  bound  separately^  and  in  cloth. 


Copyright,  1874,  by  DANIEL  W.  FISH. 


Electrotyped  by  Smith  &  McDougal,  82  Beekman  St.,  N.  Y. 


PREFACE. 


THE  author  of  this  treatise  on  Algebra  has  undertaken  the 
difficult  task  of  preparing  a  work  complete  in  one  vol- 
ume, which  shall  be  sufficiently  thorough  for  classes  in  Colleges 
and  Universities,  and  at  the  same  time  sufficiently  elementary 
for  classes  in  Common  Schools  and  Academies.  To  accomplish 
this  desirable  end  the  work  has  been  so  arranged  that  certain 
chapters  and  parts  of  chapters  may  be  omitted  by  classes  pursu- 
ing an  elementary  course. 

The  aim  has  been  :  1.  To  treat  each  subject  in  harmony  with 
the  present  modes  of  mathematical  thinking  ;  2.  To  make  every 
statement  with  such  brevity  and  precision  that  the  student  can- 
not fail  to  understand  the  meaning ;  3.  To  give  a  clear  and  rig- 
orous demonstration  of  every  proposition  ;  4.  To  present  one 
difficulty  at  a  time,  and  just  at  that  stage  of  the  student's  progress 
when  he  is  prepared  to  understand  its  treatment ;  5.  To  treat 
with  special  care  those  subjects  which  have  been  found  by  expe- 
rience to  present  peculiar  difficulties  ;  G.  To  make  the  work  thor- 
oughly practical  as  well  as  thoroughly  theoretical ;  7.  To  present 
each  subject  in  such  a  manner  as  to  create  a  love  for  the  study. 

In  the  arrangement  of  subjects  the  author  has  departed  widely 
from  the  beaten  track  ;  but  he  feels  confident  that  the  plan  he 
has  adopted  will  commend  itself  to  the  experienced  and  thought- 
ful teacher. 

To  facilitate  frequent  reviews,  ''  Synopses  for  Review  "  have 
been  placed  at  convenient  intervals  throughout  the  work. 

To  avoid  making  the  present  work  too  voluminous.  Con- 
tinued Fractions,  Reciprocal  Equations,  Elimination  by  the 
Method  of  the  Greatest  Common  Divisor,  and  Cardan's  formula 
for  cubic  equations  have  been  omitted.  These  subjects  are  treated 
in  the  Appendix  to  the  author's  "Book  of  Algebraic  Problems." 


IV  PEEFACE. 

In  preparing  the  present  treatise  the  author  has  first  consulted 
his  own  experience  as  a  teacher,  and  the  book  has  been  mainly 
written  to  meet  the  wants  of  his  own  classes ;  but  he  does  not 
hesitate  to  acknowledge  that  he  has  received  great  assistance 
from  many  sources.  A  part  of  the  material  used  in  the  chapters 
on  Positive  and  Negative  Quantities,  Greatest  Common  Divisor 
and  Least  Common  Multiple,  Fractions,  Simple  Equations,  In- 
equalities, Theory  of  Exponents,  Mathematical  Induction,  and 
the  sections  on  Permutations,  Combinations,  and  Logarithms, 
has  been  taken  from  Prof.  Todhunter's  excellent  treatise  on  Alge- 
bra. The  works  of  Bertiand,  Young,  Peacock,  Euler,  Bland, 
Goodwin,  and  Wrigley  have  been  consulted  with  advantage. 

While  the  author  has  availed  himself  of  such  material  in  the 
books  named  as  suited  his  purposes,  it  will  be  found  that  much 
of  that  so  taken  has  long  since  become  common  property,  having 
assumed  a  stereotyped  form  ;  and  that  other  portions  have  been 
very  much  modified.  It  will  be  found,  also,  that  the  present 
treatise  contains  a  large  amount  of  new  and  original  matter, 
which  has  not  been  inserted  because  it  was  novel,  but  because  it 
served  to  simplify  and  elucidate  the  subject. 

Special  attention  is  called  to  the  full  and  thorough  manner  in 
which  the  subject  of  Factoring  is  treated  ;  to  the  demonstration  of 
the  Lemma,  upon  which  the  Binomial  Theorem  depends  ;  to  the 
classification  and  treatment  of  Kadical  Quantities  ;  to  the  treat- 
ment of  Quadratic  Equations,  Higher  Equations,  Simultaneous 
Equations,  Ratio,  Proportion,  Progressions,  Interpolation,  Recur- 
ring Series,  Reversion  of  Series  ;  and  to  the  Theory  of  Equations. 

The  chapter  on  "  Logarithms  and  Exponential  Equations  "  is 
almost  entirely  the  work  of  Prof.  James  M.  Greenwood,  A.  M., 
Superintendent  of  the  Public  Schools  of  Kansas  City,  Mo.,  and 
formerly  Prof,  of  Math,  in  the  North  Missouri  State  Normal 
School;  and  the  "Synops'^s  for  Review  "have  nearly  all  been 
prepared  by  Prof.  George  S.  Bryant,  A.  M.,  of  Christian  College, 
Columbia,  Mo.  To  these  and  other  able  and  experienced  teachers 
the  author  is  also  indebted  for  many  valuable  suggestions  in  rela- 
tion to  other  portions  of  the  work. 

UXIVBRSITT  OP  THE  StATB  OP  MISSOURI,  )  THE    AUTHOB. 

Columbia,  January^  1875.  ) 


SUGGESTIONS  TO  TEACHERS. 


1.  If  the  problems  in  the  book  are  not  sufficiently  numerous 
or  sufficiently  varied,  make  some  of  your  own,  or  take  some 
from  the  book  of  "Algebraic  Problems,"  made  to  accompany 
this  volume. 

2.  The  Synopses  for  Review  should  be  placed  upon  the 
black-board,  and  dwelt  upon  until  the  topics  embraced  in  the 
review  are  thoroughly  fixed  in  the  mind  of  the  student.  To  illus- 
trate the  manner  of  conducting  a  review,  suppose  the  synopsis 
on  page  10  is  under  consideration.  Let  the  student  point  to  the 
word  "Algebra,"  and  define  it ;  then  to  "Algebraic  Quantity," 
and  define  it ;  then  to  the  two  kinds  of  Algebraic  Quantity — 
"  Known  and  Unknown" — and  define  them  ;  and  so  on. 

3.  The  following  chapters  and  parts  of  chapters  may  be 
omitted  by  classes  pursuing  an  elementary  course :  That  part 
of  Chapter  IV  from  Art.  125  to  Art.  128  inclusive,  and  from 
Art.  133  to  Art.  136  inclusive  ;  Chapter  VIII ;  Chapter  XIV ; 
that  part  of  Chapter  XVI  from  Art.  441  to  Art.  457  inclusive  ; 
Chapter  XVII ;  Arts.  482  and  483  of  Chapter  XVIII ;  all  of 
Chapter  XX  after  Geometrical  Progressions ;  Chapter  XXI ; 
Chapter  XXII ;  Chapter  XXIII. 


CONTENTS 


CHAPTER  I. 

DEFINITIONS  AND  NOTATION. 


PAGE 

DEPnnTioKS 1 

Axioms 7 


PAGB 

Notation 7 

Synopsis  yoR  Review. 10 


CHAPTER  11. 
FUNDAMENTAL  PROCESSES. 


Addition 11 

Subtraction 15 

Synopsis  for  Review ,  17 

Multiplication 18 


Synopsis  FOR  Review 26 

Division 26 

Factoring 33 

Synopsis  for  Review 40 


CHAPTER  HI. 
POSITIVE  AND  NEGATIVE    QUANTITIES. 

Relation  BETWEEN  Positive  AND  Nega-       I  A  Negative. Quantity  not  less  than 
TivE  Quantities 43  I        Zero  in  the  Arithmetical  Sense.  .  47 

CHAPTER  IV. 

GREATEST  COMMON  DIVISOR  AND  LEAST  COMMON  MULTIPLE. 


Greatest  Common  Divisor  48 

General  Rule 50 


Least  Common  Multiple. 
Synopsis  for  Review 


60 


CHAPTER  V. 

FRACTIONS. 

Definitions  and  Fundamental  Princi-       |  Combinations  op  Fractions 67 

The  Signs  of  Fractions 79 

Synopsis  for  Review 79 


PLES 61 

Reduction  of  Fractions 64 


CHAPTER   VI. 
DEFINITIONS  AND  GENERAL  PRINCIPLES  RELATING  TO  EQUATIONS. 
Definitions  and  Pbinoifles 80  I  Transformation  of  Equations 


CHAPTER  VII. 

SI^IPLE   EQUATIONS. 


Simple  Equations  with  One  Unknown 
Quantity 8i 

Simple  Equations  with  Two  Unknown 
Quantities 99 

Simple  Equations  with  any  Number  of 
Unknown  Quantities 108 


Synopsis  for  Review 120 

Discussion  of  Problems 120 

Zero  and  Infinity 126 

Finite,  Determinate,  and  Indetermi- 
nate Quantities 126 

Synopsis  for  Review 132 


CONTENTS. 


Vll 


CHAPTER  VIII. 

VANISHING    FBACTIONS — INDETERMINATE    EQUATIONS   AND    PROBLEMS — 
INCOMPATIBLE  EQUATIONS. 

Vanishing  Fractions 133  |  Incompatible  Equations 139 

Indeterminate  Equations 134    An  Impossible  Problem 141 

XjTDETKBMiNATBi  Problems 137  '  Synopsis  for  Review 141 

CHAPTER   IX. 

INEQUALITIES. 

Theorems 142  I  Equation  and  Inequality  Combined.  .145 

Solution  of  an  Inequalttt  145  I  Synopsis  for  Review 140 

CHAPTER  X. 

INVOLUTION  AND  EVOLUTION. 

Involution .147  I  Higher  Roots  of  Quantities 163 

Evolution 151  I  Synopsis  for  Review 165 


CHAPTER  XI. 

THEORY  OP  EXPONENTS. 

Basis  of  Theory  (276) 165  j  Negative  Exponents  , 

PosiTivB  Fractional  Exponents 


.167 


Synopsis  for  Review 171 


CHAPTER  XII. 

RADICAL  QUANTITIES. 


Definitions 172 

Reduction  of  Radical  Quantities  . .  173 
Combinations  of  Radical  Quantities.  183 
Involution  of  Radical  Quantities.  . .  .192 

Evolution  of  Radical  Quantities 194 

Reduction  of  Fractions  having  Surd 
Denominatobs 197 


Propositions  Relating  to  Irrational 

Quantities 803 

Simplification  of  Complex    Radical 

Quantities 206 

Imaginary  Quantities 210 

Radical  Equations 215 

Synopsis  for  Review 218 


CHAPTER  XIII. 

QUADRATIC  EQUATIONS  WITH  ONE  UNKNOWN  QUANTITY. 


Definitions  and  Principles 221 

Incomplete  Quadratic  Equations 223 

Complete  Quadratic  Equations 226 

Theory  of  Quadratic  Equations 230 


Discussion  of  the  Equation  x^  +px=q.2i'(i 

Problem  op  the  Lights 245 

Quadratic  Expressions a48 

Synopsis  for  Review ^....250 


CHAPTER  XIV. 

HIGHER  EQUATIONS  WITH  ONE  UNKNOWN   QUANTITY, 
The  Two  Forms  aa?'=c,  cKC'°+6a?'=c.... 252  |  Problems 254 

CHAPTER  XV. 

SIMULTANEOUS  EQUATIONS. 

Pairs  of  Equations  Involving  Radical 


DEFiNmoNS 255 

Pairs  op  Equations  one  of  which  is  op 
THE  First  and  the  other  of  the 

Second  Degree 255 

Particular  Systems 260 


Quantities 269 

Groups  with  more  than  Two  Unknown 

Quantities 273 

Synopsis  for  Review ^ 379 


VUl 


CONTENTS. 


CHAPTER  XVI. 
RATIO,  PROPORTION,  AND  VARIATION. 


Ratio 

Pbopobtion  . 


.280 


Variation 293 

Synopsis  fob  Review 298 


CHAPTER  XVII. 

MATHEMATICAL  INDUCTION. 
Theobems 299  |  Synopsis  roR  Review. 


.801 


CHAPTER  XVIII. 

PERMUTATIONS — COMBINATIONS— BINOMIAL   FORMULA- 
HIGHER  ROOTS. 


-EXTRACTION  OP 


Permutations 302 

Combinations 305 

BiNOMIAI.  FOBMlTIiA 307 


The  n'A  Root  of  Quantities 315 

The  n'ARootof  a  Numbee 316 

Synopsis  fob  Review 317 


CHAPTER  XIX. 

IDENTICAL  EQUATIONS. 

Pbopbrties  of  Identical  Equations.  ..318  I  Decomposition  of  Rational  Fbaction8.819 
Undetebmined  Coefficients 319  1  Synopsis  fob  Review 321 


CHAPTER  XX. 

SERIES 


GekebaIi  Definitions 822 

Abithmetical  Pboobession 323 

Arithmetical  Mean 329 

Geoicetrical  Progression 331 

Geometrical  Mean 336 

Series  BY  THE  Differential  Method. 341 
Intebpolation 844 


DEVELOPMBirr  of  Expressions  into  Se- 

BiES 846 

Recubring  Series 850 

Reversion  OF  Series ....854 

The  Binomial  Formula  for  any  Expo- 
nent  357 

Synopsis  for  Review 361 


CHAPTER  XXI. 
LOGARITHMS  AND  EXPONENTIAL  EQUATIONS. 


Logarithms 

The  Two  Principal  Systems. 


Exponential  Equations 873 

Synopsis  for  Review 374 


CHAPTER  XXII. 

COMPOUND  INTEREST  AND  ANNUITIES. 


Compound  Interest 375 

Amount  fob  nYeabs 375 


Annuities 876 

Synopsis  fob  Review 877 


CHAPTER  XXIII. 

THEORY  OF  EQUATIONS. 


Definitions 378 

General  Properties 878 

Transformation  op  Equations 389 

Theorem  of  Descartes 396 

Derived  Functions 400 


Roots  Common  to  Two  Equations 408 

Equal  Roots 403 

Limits  of  the  Roots  of  an  Equation.  .404 

Sturm's  Theorem 409 

HoRNER^s  Method  of  Approximation.. 414 


ALGEBRA. 


CHAPTER  I. 
DEFmiTIOI^S    AlsTD    NOTATIOISr. 


DEFINITIONS. 


1.  Algebra  is  that  branch  of  Mathematics  in  which  quan« 
tities  are  represented  by  letters,  or  by  a  combination  of  letters 
and  figures,  and  in  which  the  relations  of  quantities  to  each  other 
and  the  operations  to  be  performed  are  indicated  by  Signs, 

The  letters,  figures,  and  signs  are  called  Symbols, 

2.  Algebraic  Language  consists  in  the  use  of  algebraic 
symbols. 

3.  An  Algebraic  Quantity  or  Expression  is  one 

expressed  in  algebraic  language. 

There  are  two  kinds  of  algebraic  quantities — known  and 
unknown. 

4.  Known  Quantities  are  those  whose  values  are  given. 
They  are  represented  by  numbers  or  the  leading  letters  of  the 
alphabet. 

5.  Unknown  Quantities  are  those  whose  values  are  not 
given.    They  are  represented  by  the  final  letters  of  the  alphabet. 

6.  The  sign  +  is  called  the  plus  sign,  and  signifies  that 
the  quantity  to  which  it  is  prefixed  is  to  be  added.  Thus,  a  +  b 
signifies  that  b  is  to  be  added  to  a,  and  is  read  a  plus  b.  IS  a 
represents  9,  and  b  represents  3,  then  a  -{- b  is  equal  to  12. 

7.  The  Sum  is  the  result  obtained  by  addition. 


3  DEFINITIOJS'S. 

8.  The  sign  —  is  called  the  minus  sign,  and  signifies  that 
the  quantity  to  which  it  is  prefixed  is  to  be  siiUracted.  Thus, 
a  —  h  signifies  that  h  is  to  be  subtracted  from  «,  and  is  read 
a  minus  b.    If  a  is  9,  and  b  is  3,  then  a  —  bw  equal  to  6. 

9.  Oriie  Mernainder  or  I>ifference  is  the  result  ob- 
tained by  subtraction. 

10.  The  sign  x  is  called  the  sign  of  multiplication,  and 
signifies  that  the  quantity  which  precedes  it  is  to  be  multiplied 
by  the  one  which  follows  it.  Thus,  a  x  b  signifies  that  a  is  to 
be  multiplied  by  b,  and  is  read  a  multiplied  by  b,  or  a  into  b. 

The  sign  of  multiplication  is  often  omitted.  Thus,  ab  is 
equivalent  to  a  x  i.  Sometimes  a  point  is  used  instead  of  the 
sign  X .    Thus,  a'b  \s,  equivalent  to  a  x  b. 

The  sign  of  multiplication  must  not  be  omitted  when  the 
numbers  are  expressed  by  figures.  Thus,  45  is  not  equivalent  to 
4x5. 

11.  OClie  JProduct  is  the  result  obtained  by  multiplica- 
tion. 

12.  The  sign  -^  is  called  the  sign  of  division,  and  signifies 
that  the  quantity  which  precedes  it  is  to  be  divided  by  the  one 
which  follows  it  Thus,  a -r-  b  signifies  that  a  is  to  be  divided 
by  b,  and  is  read  a  divided  by  b. 

The  expression  t  is  equivalent  to  a-^b, 

13.  The  Quotient  is  the  result  obtained  by  division. 

14.  The  sign  =  is  called  the  sign  of  equality,  and  signifies 
that  the  quantities  between  which  it  is  placed  are  equal.  Thus, 
a  =  J  signifies  that  a  is  equal  to  b,  and  is  read  a  equals  b,  or 
a  is  equal  to  b. 

15.  An  Equation  consists  of  two  expressions  connected 
by  the  sign  of  equality.  Thus,  x-{-y  =  a,  a-^-b^^c  —  d,  are 
equations. 

The  First  Member  of  an  equation  is  the  quantity  on  the 
left  of  the  sign  of  equality,  and  the  Second  Member  is  the  quan- 


DEFINITIONS.  3 

tity  on  the  right  of  the  sign.  Thus,  in  the  equation,  x-}-yz=a—b, 
X  4-  y  is  the  first  member,  and  a  —  b  the  second  member. 

16.  The  sign  >  or  <  is  called  the  sign  of  inequality,  and 
signifies  that  the  quantities  between  which  it  is  placed  are  ttn- 
equal,  the  opening  being  turned  toward  the  greater.  Thus, 
ay  b  signifies  that  a  is  greater  than  b,  and  is  read  a  is  greater 
than  b ;  and  5  <  a  signifies  that  b  is  less  than  «,  and  is  read 
b  is  less  than  a. 

17.  An  Inequality  consists  of  two  expressions  connected 
by  the  sign  of  inequality,  and  its  members  are  named  as  those  of 
an  equation. 

18.  When  an  expression  is  inclosed  by  a  Parenthesis  { ), 

the  operations  which  are  indicated  in  that  expression  are  to  be 
regarded  as  performed,  and  the  parenthesis  is  to  be  regarded  as 
expressing  the  result.  Thus,  the  expression  (a  +  Z>)  (c  —  d)  indi- 
cates that  the  sum  of  a  and  b  is  to  be  multiplied  by  the  difibrence 
between  c  and  d. 

The  vinculum ,  the  brackets  [  ],  and  the  brace  \  \  hav^ 

the  same  signification  as  the  parenthesis 

Thus,      a  -\-b  X  c  —  d  \^  equivalent  to  \a  -\-b){c  —  d). 
The  vinculum   is  sometimes  placed   in   a  vertical   position. 
Thus, 

d 
is  equivalent  to  (a  +  5  —  c)  d. 


a 
—  c 


19.  The  Terms  of  an  expression  are  the  parts  which  are 
connected  by  the  sign  +  or  the  sign  — .  Thus,  a,  b,  c,  and  d 
are  the  terms  of  the  expression  a  -{■  b  —  c  -\-  d. 

20.  A  Polynomial  is  an  expression  containing  two  or 
more  terms. 

21.  A  Binoinial  is  a  polynomial  containing  only  two 
terms.    Thus,  abc  -f  a;  is  a  binomial. 

22.  A  Trinomial  is  a  polynomial  containing  only  three 
terms.    Thus,  ab  •{-  ac  --  be  is  a  trinomial 


4  DEFINITIONS. 

23.  A  31onoinial  is  an  expression  which  does  not  contain 
parts  connected  by  the  sign  +  or  the  sign  — .  Thus,  abc  is  a 
monomial. 

24.  When  one  quantity  is  the  product  of  two  or  more  other 
quantities,  each  of  the  latter  is  called  a  Factor  of  the  product. 
Thus,  fl,  h,  and  c  are  factors  of  the  product  ahc. 

25.  A  Numerical  Factor  is  one  which  is  expressed  by 
a  figure,  or  figures. 

26.  ^  Literal  Factor  is  one  which  is  expressed  by  a 
letter,  or  letters. 

27.  When  a  product  contains  one  factor  which  is  numerical, 
and  another  which  is  literal,  the  former  factor  is  called  the 
Coefficient  of  the  latter.  Thus,  in  the  product  'tabc,  7  is  the 
coefficient  of  dbc. 

The  term  coefficient  is  sometimes  applied  to  any  factor. 
Thus,  in  the  product  '7 abc,  7«  may  be  called  the  coefficient  of 
he,  and  in  the  product  ahc,  a  may  be  called  the  coefficient  of  be, 
or  ab  the  coefficient  of  c. 

28.  A  Foiver  of  a  quantity  is  the  product  of  factors  each 
of  which  is  equal  to  that  quantity.  Thus,  «  x  «  is  the  second 
power  of  a;  «  x  «  X  «  is  the  third  power  of  a;  a  x  a  x  a  x  a 
is  the  fourth  power  of  a  ;  and  so  on. 

The  first  power  of  a  is  a. 

29.  An  Fxponent  is  a  number  placed  on  the  right  of, 
and  a  little  above  a  quantity,  and  indicates  how  many  times  the 
quantity  is  to  be  used  as  a  factor.  Thus,  a^  is  equivalent  to 
^  X  a\  a^  is  equivalent  to  a  x  a  x  a',  «*  is  equivalent  to 
T,xaxaxa\  and  so  on.  If  no  exponent  is  expressed,  1  is 
understood.     Thus,  a  is  equivalent  to  a\ 

The  product  of  n  factors  each  equal  to  a  is  expressed  by  a", 
and  n  is  called  the  exponent  of  a. 

The  second  power  of  a,  that  is,  a^,  is  often  called  the  square 
of  a,  and  the  third  power  of  a,  that  is,  a^,  is  often  called  the 


DEFINITIONS.  0 

cube  of  a.    The  expression  a^  is  read  a  to  the  fourth  power,  or 
t)riefly,  a  to  the  fourth  ;  and  a"  is  read  a  to  the  n^\ 

30.  The  Square  Hoot  of  any  given  quantity  is  that 
quantity  which  has  the  given  quantity  for  its  square  or  second 
potoer  ;  the  cuhe  root  is  that  which  has  the  given  quantity  for  its 
cule  or  third  power  ;  ihQ  fourth  root  is  that  which  has  the  given 
quantity  for  ii^  fourth  power  ;  the  fifth  root  is  that  which  has  the 
given  quantity  for  its  fifth  power  ;  and  so  on. 

31.  The  Hadical  Sign  ^  indicates  that  some  root  of 
the  quantity  to  which  it  is  prefixed  is  to  be  found. 

33.  The  Index  of  the  root  is  the  number  placed  above  the 
radical  sign. 

The  square  root  of  a  is  denoted  thus,  ^«,  or  simply  thus, 
j^a ;  the  cube  root  of  a  is  denoted  thus,  ^a ;  the  fourth  root  of  a 
is  denoted  thus,  ^a ;  and  so  on. 

33.  The  symbols  employed  in  Algebra  are  classified  as  fol- 
lows: 

1.  Symbols  of  Quantity  are  letters  and  other  characters 
used  to  represent  quantities. 

2.  Symbols  of  Operation  are  the  signs  +,  — ,  x,  -i-, 
/y/,  and  the  exponential  sign. 

3.  Symbols  of  Helation  are  the  signs  =,  >,  <,  and 
others  to  be  explained  hereafter. 

4.  Symbols  of  Aggregation  are  the  signs  (  ),  [  ], 
I   i,  — ,  and  |. 

34.  Similar  or  Like  Quantities  are  those  whose  lit' 
eral  parts  are  identical.  Thus,  ^ab,  lOab,  12ab,  and  'Zbah  are 
similar. 

35.  I>i8similar  or  Unlike  Quantities  are  those  whose 
literal  parts  are  different.     Thus,  4:ab  and  lOa^  are  dissimilar. 

Remark. — An  exception  must  be  made  in  those  cases  where  letters  are 
considered  as  coefficients.  Thus,  Oic^  and  bj^  are  similar  if  a  and  b  are  con- 
sidered as  coefficients. 


6  DEFINITIONS. 

36.  Each  of  the  hteral  factors  of  a  term  is  called  a  Dinien- 
8 ion  of  the  term,  and  the  Def/7*ee  of  a  term  is  equal  to  the 
number  of  its  dimensions.  The  degree  of  a  term,  therefore,  is 
equal  to  the  sum  of  the  exponents  of  its  literal  factors.  Thus, 
6aWc  is  of  the  sixth  degree. 

37.  A  Homogeneous  Polynomial  is  one  whose  terms 
are  all  of  the  same  degree.  Thus,  7a^  +  Za?h  +  ^^ahc  is  homo- 
geneous. 

38.  Hie  Numerical  Value  of  an  algebraic  expression 
is  the  number  obtained  by  substituting  for  each  letter  its  numeri- 
cal value,  and  then  performing  the  indicated  operations.  Thus, 
if  a  =  5  and  J  =  6,  the  numeiical  value  of  the  expression 
3a  —  U  is  3. 

Some  terms  of  frequent  use  in  Algebra  are  here  defined. 

39.  A  JProposition  is  the  statement  of  a  truth,  or  of 
something  to  be  done. 

Propositions  are  of  the  following  kinds:  Axioms,  theorems, 
lemmas,  problems,  postulates,  corollaries,  scholiums. 

1.  An  Axiom  is  a  self-evident  truth. 

2.  A  TJieorem  is  a  truth  requiring  demonstration. 

3.  A  Lemma  is  an  auxiUary  theorem  used  in  the  demonstra- 
tion of  another  theorem. 

4.  A  Problem  is  a  question  proposed  for  solution. 

0.  A  Postulate  assumes  the  possibility  of  the  solution  of  some 
problem. 

6.  A  Corollary  is  an  obvious  consequence  deduced  from  one 
or  more  propositions. 

7.  A  SchoUtim  is  a  remark  upon  one  or  more  propositions. 

40.  An  Hypothesis  is  a  supposition,  made  either  in  the 
enunciation  of  a  proposition,  or  in  the  course  of  a  demonstra- 
tion. 

41.  A  H^ormula  is  a  theorem  expressed  in  algebraic  lan- 
guage. 


NOTATION^. 


42.  AXIOMS. 

1.  The  whole  is  equal  to  the  sum  of  all  its  parts. 

2.  If  equal  quantities  be  added  to  equal  quantities,  the  sums 
will  be  equal. 

3.  If  equal  quantities  be  subtracted  from  equal  quantities,  the 
remainders  will  be  equal. 

4.  If  equal  quantities  be  multiplied  by  the  same  or  by  equal 
quantities,  the  products  will  be  equal. 

5.  If  equal  quantities  be  divided  by  the  same  or  by  equal 
quantities,  the  quotients  will  be  equal. 

6.  Quantities  that  are  equal  to  the  same  quantity  are  equal  to 
each  other. 

NOTATION. 

43.  Algebraic  Notation  consists  in  representing  quan- 
tities, operations,  and  relations  by  means  of  symbols. 


EXAMPLES    IX    NOTATION. 

44.  Express,  in  algebraic  language,  the  following  eight  state- 
ments : 

1.  The  second  power  of  a,  increased  by  twice  the  product  of 
b  and  c,  diminished  by  the  second  power  of  c,  and  increased  by  d, 
is  equal  to  m  times  x.  Ans,  a^  +  2bc  —  r?  ■\-  d  =  mx. 

2.  The  quotient  arising  from  dividing  a  by  the  sum  of  %  and 
h,  is  equal  to  twice  h  diminished  by  c. 

Ans.  7  =  25  — -  c. 

X  -{-  b 

3.  One-third  of  the  remainder  obtained  by  subtracting  four 

from  six  times  x,  is  equal  to  the  quotient  arising  from  dividing 

five  by  the  sum  of  a  and  b.  ^       6a;  —  4  5 

•^  Ans.  — - —  — 


a-\-b 


8  NOTATION. 

4.  Three-fourths  of  the  sum  of  x  and  five,  is  equal  to  three- 
sevenths  of  l,  diminished  by  seventeen. 

Am,  I  (a:  +  5)  =  ^5  -  17. 

5.  One-ninth  of  the  sum  of  three  times  x  and  J,  added  to  one- 
third  of  the  sum  of  twice  x  and  four,  is  equal  to  the  product  of 
fl,  J,  and  c,  ^^^  1  (2^;  +  4)  +  ^  (So:  +  J)  =  abc, 

6.  The  quotient  arising  from  dividing  the  sum  of  a  and  h  by 
the  product  of  c  and  d,  is  greater  than  p  times  the  sum  of  m,  n,  x, 

and  y*  -«       ^  +  ^  ^     /      .       .       .     \ 

^  Ans.  — -j—  >  p{m  -\-  n  -\-  X  -^^  y). 

7.  The  square  root  of  the  sum  of  a  and  h  is  equal  to  m  times 
the  cube  root  of  the  remainder  obtained  by  subtracting  y  from  x, 

Ans.  ^/a  ■\-  h  =.  m'^x  —  y. 

8.  The  square  root  of  x,  diminished  by  the  square  root  of  y, 
is  equal  to  n  times  the  sum  of  the  fourth  root  of  a  and  the  fourth 
root  of  h.  Ans.  ^x  —  ^y  —  n  {%/a  +  yh). 

45.  Express,  in  common  language,  the  following  six  algebraic 
expressions : 

^  a  -{■  X      X m 


b  c~  a  -\-  b' 

Ans.  The  quotient  arising  from  dividing  the  sum  of  a  and  x 
by  b,  increased  by  the  quotient  of  x  divided  by  c,  is  equal  to  the 
quotient  of  m  divided  by  the  sum  of  a  and  b. 

2.  3a2  -^  [^  —  c){d-{-e)=x  —  y. 

3.  3a^  +  b  —  c{d-{-e)=x  —  y. 

a  -\-  X  a  —  y  _     m 


5H-d  +  c  3  a  +  b 

^  Vdb-h^Vc      .     ,    m 

0.  —777 =z  ox  -\ . 

a -{-2b  n 


NOTATION.  ^ 


46.  NUMEKICAL    VALUES. 

If  «  =  1,   b  =  3,  cz=4:,  d  =  6,   e  =  2,   and  /=  0,  find  the 
numerical  yalue  of  each  of  the  following  ten  expressions : 

1.  a  +  2b  +  4c. 

2.  3b  ^6d  —  2e. 

3.  ab  +  2bo  +  Sed. 

4.  ac  +  4:cd  —  2 Je. 

5.  abc  +  45^  +  ec  —  ^. 

6.  ^2  +  J2  ^  c2  +/2. 

r,   cd       4:be       cd 
T  "^  'S^"  ~  24* 

8.  c4  —  4c8  +  3c  —  6. 

9     ^  +  ^^ 
*  2c  —  3a 

10.  ^(27b)  -  V(2c)  -f  V(2c). 

11.  Find  the  value  of  (x  -{-  y){x^  y),  when  a;  =  8  and  y  =  5. 

Ans.  39. 

12.  Find  the  value  of  x  +  y  x  x  ^  y,  when  a;  =  8  and  t/  =  5. 

^W5.  43. 

13.  Find   the  value  of   x  ^{x^  —  Sy)  -{-  y  ^{x^  +  Sy)y  when 
X  =  5  and  y  =  8.  Ans.  26. 

14.  Find  the  value  of  (b  —  x)  {^a  +  &)  +  ^\(a—b)  {x^-y)\, 
when  a  =  16, 5  =  10,  xz=z  5,  and  ^  =  1.  ^^5.  76. 


^7^5. 

23. 

-47i5. 

35. 

Ans. 

63. 

Ans. 

88. 

Ans. 

92. 

Ans. 

26. 

Ans. 

15. 

Ans. 

6. 

Ans. 

5. 

Ans. 

9. 

15.  Find  the  value  of  a;  in  the  equation  x  = ^ 

when  a  =  10,  5  =  5,  c  =  4,  c?  =  2,  and  6  =  3. 


Ans.  X  =  20. 


10 


DEFINITIONS    AND    NOTATION. 


47. 


SYNOPSIS    FOR    REVIEW. 


Algebba. 


Algebraic  Quantity.  \  ^"<^^' 

(   Unknown. 


Sum.    Remainder.    Product.    Quotient. 
Equation | 

Inequality j 


Firet  Member. 
Second  Member. 

First  Member. 
Second  Member. 


Names  op 
Expressions. 


o 

EH 

< 

Eh 

o 

HH  iz; 

eg 


Terrm. 
Polynomial 


Symbols 


Binomial. 

Trinomial. 
Monomicd  .  .  . 

Numerical. 
Factor \  Literal. 

CoeflBcient. 

Known :  a,  b,  c,  d,  etc. 
'  Of  quantity  •  .  .  ■(  Unknown :    t,  u,  v,  w, 
X,  y,  z. 

{"T  i  ?      X,    .J    "T-y  "71 

V    ,  Exponent. 
Of  relation, .  .  .  |  =,  >,  <. 

.  Of  aggregation.  |  (  ),    H  j  [  ]  ^  — >  I  • 
Similar  Quantities.    Dissimilar  Quantities. 
Degree  of  a  Term.    Homogeneous  Polynomial. 
Numerical  Value. 


Proposition 


Axiom. 
Theorem. 
Problem. 
Postulate. 
CoroUary. 
L  Scholium. 


Hypothesis.    Formula. 


CHAPTEE   II. 
FUNDAMENTAL    PROCESSES. 


ADDITION 

48.  Addition  is  the  process  of  finding  the  simplest  expres- 
sion for  the  sum  of  two  or  more  quantities. 

49.  A  JPositive  Term  is  one  which  is  preceded  by  the 
^gn  + .  When  a  term  has  no  sign  prefixed,  the  sign  +  is  un- 
derstood. 

50.  A  Negative  Term  is  one  which  is  preceded  by  the 

sign  — . 

ORDER    OF    TERMS. 

51.  The  value  of  a  polynomial  whose  terms  are  positive  is 
the  same  in  whatever  order  the  terms  may  be  written.  Thus, 
a-\-h-\-c=a-\-c-\-h=h-\-c-{-a=c-\-a-\-b=c-\-h-\-a=l)-\-a-^c. 

53.  When  a  polynomial  contains  both  positive  and  negative 
terms,  we  may  write  the  former  tenns  first  in  any  order,  and  the 
latter  after  them  in  any  order.  Thus,  a-\-b—c—e=za-\-h—e—c 
=b-i-a — c — e=b-\-a — e — c. 

53.  In  some  cases  we  may  vary  the  order  of  terms  still  further. 
Thus,  if  a  =  10,b  =  6,  and  c  =  o,  then  a  -\-  b  —  c  =  a  —  c  -{-  b 
z=:b  —  c  •\-  a. 

But,  if  a  =  2,  b  =  Q,  and  c  =  b,  the  expression  a  —  c  -{- h 
presents  a  difficulty,  because  we  are  apparently  required  to  sub- 
tract 5  from  2.  It  will  be  convenient  to  agree  that  such  an  ex- 
pression as  a  —  c  -\-by  when  c  is  greater  than  a,  shall  be  under- 
stood to  be  equivalent  to  a  -\-  b  —  c.  At  present  we  shall  not 
use  such  an  expression  except  when  c  is  less  than  a  -\-  b. 

In  like  manner  we  shall  consider  —b-\-a  as  equivalent  to 
a  —  b.    We  shall  recur  to  this  point  in  Chapter  III. 


12  FUNDAMENTAL    PROCESSES. 


REDUCTION    OF    SIMILAR    TERMS. 

54,  When  two  or  more  terms  of  a  polynomial  are  similar,  it 
may  be  reduced  to  a  simpler  form. 

1.  Let  it  be  required  to  simplify  the  expression  ^a%  —  Sa^c 
+  9ac2  —  %a^b  +  "id'c  —  6R 

This  expression  may  be  written  thus :    4a2J  —  2a2J  +  la^c 

—  Sa^c  +  9flc2  —  6^  (53).     Now  Aa^b  —  2a^  =  2a%  and   ^a^c 

—  dah  =  4:0.^6.    Hence  the  given  expression  may  be  reduced  to 
2a^b  +  4a2c  +  Qac^  —  6&2. 

2.  Let  it  be  required  to  simplify  the  expression  2a^b(^  —  ^a^b(^ 
+  Wbc^  —  8a3^>c2  4-  \\a%(?. 

We  write  the  positive  terms  in  one  column  and  the  negative 
terms  in  another  thus, 

%a^b(?  -  ^a%& 


19a3^>c2  __  V2a^b(^  =  Wbc\ 


3.  Let  it  be  required  to  simplify  the  expression  ^abc  -f  Sa^J 
+  2a^  —  ha^b  —  Za^b. 

Arranging  the  terms  thus, 

4:abc  +  3fl2J  —  5^25 

-f  'ZaJ^b  —  Sa^b    and  uniting, 
we  obtain  Aabc  -f  5a^  —  Sa^. 

But^bc  +  6a^—Sa^=^abc+5a^—5a^—da^z=4abc—Sa^, 


RULE. 

L  Reduce  the  positive  similar  terms  to  one  term  by  addition, 

n.  In  like  manner  reduce  the  negative  similar  terms  to  one 
term. 

in.  Then  subtract  the  less  result  from  the  greater,  and  to  the 
remainder  prefix  the  sign  of  the  greater. 


ADDITION.  13 

EXAMPLES. 

Keduce  each  of  the  following  expressions  to  its  simplest  form : 

1.  10^4  -f  3a*  +  6a*  —  a*  —  5a*.  Ans.  UaK 

2.  ba^b  +  3  ^/aFi  —  "tab  +  17a5  +  2  V~d^  —  Wb  —  SV~a¥c 
—  10«^  +  9«'^-  ^ns.  Sa^b  —  3  V^c. 

3.  3a  —  2a  —  7c  +  3c  +  2a  +  4c  —  3a.  Ans.  0. 

4.  %h  —  8ac2  +  15J3c  +  8ac  +  9ac2  —  24Z>8c. 

Ans.  ac?  +  8ac. 
6.  6ac2  —  bab^  +  7ac2  —  ^a¥  —  13ac2  -f-  Uab\    Ans.  lOal^. 
6.  d^b  —  dab^-{-  Sa^  +  5c  —  Sa^b  +  8a^+  2a^  +c  +ab^—Sc. 
55,  To  find  the  sum  of  two  or  more  quantities. 

1.  Let  it  be  required  to  find  the  sum  of  c  —  d  -{-  e  and  a  —  b. 
Adding  c  -\-  e  to  a  —  J,  we  obtain  a  —  b-\-c  +  e',  we  have, 

however,  added  d  too  much  to  a  —  b\  hence,  in  order  to  obtain 
the  correct  sum,  d  must  be  subtracted  from  a  —  b-{-c-\-e.  We 
thus  obtain  a  —  b-\-c-\-e  —  d.  Therefore,  to  find  the  sura  of 
quantities,  all  the  terms  of  which  are  unlike,  write  them  in  any 
order,  prefixing  to  each  term  its  proper  sign. 

2.  Let  it  be  required  to  find  the  sum  of 

aS  +  3a2  —  4aJ 

2a2  —  3aJ  +    J2  —    c, 
and  a^  +  2a5  —  bb'^  +  3c.    Their  sum,  after 

reducing,  is  a^  +  ^a?  —  bab  —  46^  ^  2c    (54). 

BZTLE. 

I.  Write  similar  terms,  with  their  proper  sigfis,  in  the  same 
colum7i. 

II.  Reduce  each  column  (54),  and  to  the  results  annex  those 
terms  which  cannot  be  reduced,  prefixing  to  each  its  proper  sign. 

EXAMPLES. 

(1.)  (2.) 

7a;  4_  Sab  +  2c  IQa^^  -f     be  —  2abc 

—  Sx  —  Sab  —  be  —    ^aW  +  %c  +  Qabc 

bx  —  "dab  4-  9c  —  lOa^^  —  12Z>c  +  3a^c 

Sum         9a;  —  9a5  +  6c.  2a^l^  —  2bc  +  '^abc. 


14  FUNDAMENTAL    PROCESSES. 

3.  Find  the  sum  of  4:a  —  ob  -\-  Sc  —  2dj  a  -^  b  —  4:C  +  bd, 
3a  —  7J  +  6c  +  46?,  and  a-\-  ^b  —  c  —  Id.     Ans.  9«  —  75  +  4c. 

4.  Find  the  sum  of  a^  ^  2^^  —  3x -^  1,  2a:^  —  3x^  +  4a;  —  2. 
dx^  -{-^  •\-  5,  and  ^  —  dx^  —  5x  +  d.      Aris.  lOa:^  —  4a;  +  13. 

5.  Find  the  sum  of  x^  —  dxy  -{-  y^  -\-  x  -\-  y  —  1,  2x^  -{■  4:xy— 
'Sy2  —  2x  —  2y  +  3,  3a;2  _  5^^  _  4y2  ^^x  +  4:y  —  2,  and  6x^  + 
lOxy  -\-  5y^  -j-  X  -\-  y.  Ans.  122^^4-  Gxy  —  y^  +  Zx  -f  4?/. 

6.  Find  the  sum  of  a;^  —  2a3?  +  a%  a?  +  3ffa;2^  and  27^  —  ax^. 

A71S.  ^  +  aH, 

7.  Find  the  sum  of  4aJ  —  ar^^  Sar^  —  2aJ,  and  2aa;  +  25a;. 

Ans.  2ab  +  2a;2  +  2aa;  +  25a;. 

8.  Find  the  snm  oi  a  +  b  -\-  c  -\-  dy  a  -\-  b  ■}-  c  —  d,  a  -\-b  — 
c  +  d^  a  —  b  -{-  c  -\-  df  and  — a-i-b  +  c-\-d. 

Ans.  3a  +  35  +  3c  +  dd. 

9.  Find  the  sum  of  3  (a?»  —  y%  8  {x^  —  y^),  —  5  (a;2  —  y^)  -f 
6  (a;2  ^  y2)2,  and  7  (a;  —  y)2. 

Ans.  6  (a;2  _  y2)  ^  6  (a;2  +  7/2)2  +  7  (a;  -  ;?/)2. 

10.  Find  the  sum  of  a"*  —  5''  +  3a: ",  205"*  —  35"  —  x^,  and 
or  4-  45**  —  a;«.  Ans.  4a"*  +  2a;^  —  xf^. 


LAWS  RESPECTING  THE  USE  OF  THE  PARENTHESIS. 

56,  If  a  polynomial  has  any  number  of  its  terms  enclosed  by  a 
parenthesis  preceded  by  the  sign  + ,  the  parenthesis  may  be  omitted 
and  the  value  of  the  polynomial  will  not  be  changed.     Thus, 

a  —  b  +  {c  —  d-\-e)=a  —  b-\-c  —  d  +  e  {^5). 

Cor. — The  value  of  a  polynomial  will  not  be  changed  by 
enclosing  any  number  of  its  terms  by  a  parenthesis^  provided  the 
parenthesis  have  the  sign  +  prefixed.    Thus, 

a  —  5  +  c  —  d-\-  e=za  —  b  ■\-  c-^  {—d  -\-  e)=. 
a  —  d  +  (c  -\-  e—b)  =  a  -^  {—  d -\-  c  ■{■  e  —5). 


SUBTRACTION.  16 


SUBTRACTION. 


11 
5-3 


6  +  3=9. 


57.  Subtraction  is  the  process  of  finding  the  simplest  ex^ 
pressioQ  for  the  difference  between  two  quantities. 

5S,  Uie  Minuend  is  the  quantity  from  which  the  sub- 
traction is  to  be  made. 

59.  TTie  Subtrahend  is  the  quantity  to  be  subtracted. 

60.  The  Hefnainder  is  the  result  obtained  by  the  sub- 
traction. 

61.  To  find  the  diflference  between  two  quantities. 

1.  Let  it  be  required  to  subtract  5  —  3  from  11. 
Subtracting  5  from  11  we  obtain  6.    This 

result  is  too  small  by  3,  for  the  number  5  is 
larger  by  3  than  the  number  which  was  required 
to  be  subtracted.  In  order,  therefore,  to  obtain 
the  correct  remainder,  3  must  be  added  to  6. 

2.  Let  it  be  required  to  subtract  c  —  d  from  a  -\-  b. 
Subtracting   c   from    a  +  5    we  obtain 

a  ^  h  —c.    This  result  is  too  small  by  d, 

for  c  is  larger  by  d  than  the  quantity  which 

was  required  to  be  subtracted.    In  order, 

therefore,  to  obtain  the  correct  remainder,  d  must  be  added  to 

a  +  b  —  c.     Hence  a-{-b  — {c  —  d)  =  a-{-I?—c-}-d. 

The  same  result  may  be  obtained  by  adding  —  c  +  ^  to 
a  +  b  (55). 

RULE. 

CJiange  the  sign  of  every  term  of  the  subtrahend  from.  -\-  to  —y 
or  from  —  to  •\-,  and  add  tUe  result  to  the  minuend. 

Remakks— 1.  Beginners  may  solve  a  few  examples  by  actuaUy  changing 
the  sign  of  every  term  in  the  subtrahend.  After  this,  it  is  better  merely  to 
conceive  such  change  to  be  made. 

2.  Subtraction,  in  Algebra,  is  proved  in  the  same  manner  as  in  Arith- 
metic, by  adding  the  remainder  to  the  subtrahend ;  the  sum  should  be  equal 
to  the  minuend. 


a  +  b 
c  —  d 


a  -\-  b  —  c  -\-d. 


Id  FUNDAMENTAL    PROCESSES. 

EXAMPLES, 

(1.) 

From  3a  +    J  ^  f     3a  +    5 

Take  a-    J     ^^^  ^°^^  ^^^^  *^®  ^^^^^  ""^{-a^-    I 

T^       .    -,       7 ^         the  subtrahend  chaQged.     —r rr 

Remainder    2a  +  2J  J  ^       I    2a  +  25 

(2.)  (3.) 

From  \\a^-\-Zdb^^x\j  From  5«_3J  +  4c— 7rf 

Take  5a'^  +  4a^>— 6a;y  Take  2a— 2^>  +  3c—  d 

Remainder    Ga^—  aj-|-2a;^.  Remainder    3a—  5+  c— 6^. 

4.  From  7^-\-^3?  — Ix^-^-'^x  — \  take  a:*  +  2a;3  —  2^:2  + 
6a;  —  1.  Ans,  2a^  +  a;. 

5.  From  3a2  —  2ax-{-a^  take  a^^ax+  x\  Ans.  2a^  —  ax. 

6.  From  2a  —  2J  —  c  +  ^  take  a  —  b  —  2c+  2d, 

7.  From  4a  +  35  —  2c  +  Sd  take  a  +  25  +  c  —  5d 

J?iA'.  3a  +  5  — 3c  +  13d, 

8.  From  Sar^  _  3aa:  +  5  take  5x^  +  2aa:  +  5. 

9.  From  1!xy  —  10^  +  4a;  take  3a:y  +  3y  +  3a;. 

^W5.  4:xy  —  ISy  +  a;. 

10.  From  3a  4-  5  +  c  take  a  —  5  —  c.     Ans.  2a  +  2b  -\-  2c. 

11.  From  a2  -|-  2a5  +  5^  take  a^  —  2a5  +  52.  v4m5.  4a5. 

12.  From  a^  +  352c  +  a52  —  abc  take  a52  —  fl5c  +  b\ 

13.  From  5a;2  —  y^  take  4a:2  _  ^  ^  ^4  ^^^^  2;2  —  y*. 

14.  From  4a'"+  2xp—  x^  take  a*"—  5'»+3a;i'+  2a"»—  35"— a;**. 

^W5.  a"*+  45"— a:'. 

LAWS   RESPECTING  THE  USE  OF  THE   PARENTHESIS. 

63,  A  parenthesis  preceded  by  the  sign  —  may  be  omitted 
without  affecting  the  value  of  the  expression  in  which  it  occurs, 
provided  the  sig7i  of  every  term  within  it  be  changed.    Thus, 
a—{b  —  c^d-\-e)=^a  —  b-\-c-\-d  —  e. 

Cor. — The  value  of  an  expression  will  not  be  changed  by  en- 
closing  any  number  of  its  terms  by  a  parenthesis,  preceded  by  the 
sign  — ,  provided  the  sign  of  every  term  thus  enclosed  be  changed. 
Thus, 

a—b-{-c-\-d—ez=za—b-\-c—{—d-\-e)=a—{b—c—d-\-e)  = 

a  +  c— (5— £?+e). 


SUBTRACTIOK. 


17 


EXAMPLES. 


Ans.  a  —b  +  c  —  d. 
Ans.  a  —  h  -{•  c  -\-  d* 
Ans.  a  —  7 J. 
Ans,  5a, 


Eemove  the  signs  of  aggregation  from  each  of  the  following 
expressions : 

1.    a—[b—(c  —  d)]. 
%    a-  [(5  —  c)  —  d]. 

3.  «  4-  25  —  6«  —  [U  —  (Qa  —  65)]. 

4.  7fl  —  f  3a  —  [4fl  —  (5a  —  2fl)]  i . 

5.  da—\a  +  h—  \a^h-\-c—{a  +  h  -\- c  +  d)W 

Ans.  2a  —  h^d. 

6.  2x _  [3y  —  {4a;  —  (by  —  6x)\].  Ans.  12a;  —  Sy. 

7.  a  —  {a  —  [a  —  (a  —  x)]\.  Ans.  x. 

8.  4a;  —  I  a;  —  [a:  —  (a;  —  3)  +  3]  —  3  }  —  {  —  a;  —  [— a; 
(_  a;  +  3)  4-  3]  _>  3}.  Ans.  4a;  +  12. 


63. 


SYNOPSIS    FOR    REVIEW 
Positive  terms. 
Negative  terms. 

Order  of  terTns. 


ADDmON. 


j  When  preceded  by  + . 
( When  preceded  by  — . 


SUBTBACnON.  " 


Use  op 
Pahknthesis. 


Reduction  of  terms.  1^^^^  P««^^^^ 
•^  (Rule. 

Sum. 

Mule. 

JUimiend. 

SvbtraJiend. 

Remainder. 

Rule. 

Toremcyte (When  preceded  by +. 

( When  preceded  by  — , 

To  introduce.  .    .    .\  ^^^^  P^^^^^^^  ^^  +* 
( When  preceded  by  — . 


18  FUNDAMENTAL    PROCESSES. 


MULTIPLICATION. 

64.  TJie  ^Product  of  two  quantities  is  a  quantity  which  is 
as  many  times  greater  than  one  of  them  as  the  other  is  greater 
than  a  unit    Thus,  the  product  of  5  and  4  is  20. 

65.  Multiplication  is  the  process  of  finding  the  product 
of  two  quantities. 

66.  The  Multiplicand  is  the  quantity  which  is  to  be 
multiphed. 

67.  The  Multiplier  is  the  quantity  by  which  to  multiply. 

68.  The  product  of  three  or  more  quantities  is  sometimes 
called  a  Continued  Product 

The  product  of  any  number  of  factors  is  the  same  in  whatever 
order  they  may  be  taken ;  thus,  abc  =  ach  =  bca.  The  literal 
factors  are  generally  arranged  in  alphabetical  order. 

69.  To  find  the  Product  of  Monomials. 

Let  it  be  required  to  find  the  product  of  taW  and  ha^¥c» 
We  may  indicate  the  multipHcation  thus : 

laW  X  6a^¥c\ 

and  since  the  product  is  the  same  in  whatever  order  the  factors 
are  taken,  we  have 

la^J^  X  ba%^c  =  7  x  5  x  a^aW¥c  =  dhaHW¥c. 

Here  a  occurs  as  a  factor  five  times,  h  occurs  six  times,  and  c 
once.    Therefore  the  required  product  may  be  written  thus : 

ZbaWc.     Hence, 

Principles. — 1.  The  coefficient  of  the  product  of  given  mono- 
mials is  equal  to  the  product  of  their  coefficients. 

2.  Every  letter  which  occurs  in  any  of  the  given  factors  must 
he  written  in  the  product  with  an  exponent  equal  to  the  sum  of 
all  its  exponents  in  the  given  factors. 


MULTIPLICATION. 


19 


Cor. — We  may,  if  we  please,  indicate  the  product  of  the  hke 
powers  of  different  letters  by  writing  them  within  a  parenthesis 
and  placing  the  exponent  over  the  whole.    Thus, 

aW  =  {aVf ;  for  {aUf  =  ahxah  =  aaxhh  =aW. 


EXAMPLES. 


1.  Multiply  ah  by  x. 

2.  Multiply  ^ax  by  2ay. 

3.  Multiply  4am  by  Zlc^n. 

4.  Multiply  ba^x  by  Zah?. 

5.  Multiply  Za^x""  by  ^a'^oif*. 


Ans.  abx, 
Ans.  ^dhoy. 
Ans.  VZdbc^mn, 
Ans.  15aV. 
Ans.  27a'"+»a;'"+'» 


70.  To  find  the   Product  of  two  Polynomials. 


1.  Let  it  be  required  to  multiply  a  -f  5  by  «?.  The 
product  of  a  and  c  is  ac ;  but  this  is  too  small  by  he, 
for  it  is  the  sum  of  a  and  h  which  is  to  be  multiplied 
by  c.    Hence 

{a  +  h)c  ^  ac  -\-  he. 

2.  Let  it  be  required  to  multiply  a  —  hhy  c. 
Here  the  product  of  a  and  c  must  be  diminished 

by  the  product  of  h  and  c.    Hence 

{a  —  h)c  =  ac  —  he. 

3.  Let  it  be  required  to  multiply  a  -{-  b  by  c  ■{■  d. 
The  product  of  a  -^  h  and  e  is  ac  -}-  he; 

but  a  -f  &  is  to  be  multiplied  by  the  sum 
of  c  and  d ;  hence  ae  4-  he  is  too  small 
by  the  product  of  a  -\- h  and  d ;  that  is, 
by  ad  +  hd,  which  must,  therefore,  be 


a-{-b 
e-\-  d 


a  +  b 


ae  -f  be. 


a  - 

-b 

c 

ae- 

-be. 

ae  +  he  +  ad  -\-  bd. 
added  to  ae  +  he  to  produce  the  correct  result.    Hence 


(a  -^  h)  {e  +  d)  =  ac  -^  be  -\-  ad  -{-  bd. 


*0  FUNDAMEJSTAL    PROCESSES. 

4.  Let  it  be  required  to  multiply  a  -^  bhy  c  —  d. 
Here  the  product  of 


a  +  b  and  c  must  be  di- 
minished  by  the  product 
ota  +  h  and  d.    Hence 


c—d 


ac  +  bc—(ad-{-bd)=ac  +  bc—ad—M. 
{a  -\-b)  (c  —  d)  =  ac  -{-be—  {ad  +  bd)  =zac-^bc  —  ad  —  bd, 

5.  Let  it  be  required  to  multiply  a  —  bhj  c  —  d. 
Here  the  product  of 


a  —  b  and  c,  which  is 
ac—bc,  is  to  be  dimin- 
ished by  the  product 
of  a  —  b  and  d,  which 
ha  ad  —  bd.    Hence 


a — b 
c—d 


ac—bc— {ad— M)^aC'-bc — ad-^bd. 


{a  —  b)  {c  —  d)  =  ac  —  bc  —  {ad  —  bd)  =  ac  —  bc  —  ad-\-  bd. 

In  this  example,  we  observe  that  corresponding  to  the  terms 
—  b  and  c,  one  of  which  occurs  in  the  multiplicand  and  the  other 
in  the  multiplier,  there  is  the  term  —  be  in  the  product ;  and 
corresponding  to  —  J  of  the  multiplicand  and  —  d  of  the  multi- 
pher,  there  is  the  term  -j-  bd  in  the  product.  Hence  it  is  often 
stated  as  an  independent  truth,  that 

(—b)  X  c=  —  be,  and  (—  b)  x  {—  d)  =  ■{-  bd. 

Thus,  the  sign  of  the  product  is  deduced  from  the  signs  of  the 
factors  by  the  rule. 

Like  signs  produce  +,  and  unlike  signs  produce  — . 

6.  Let  it  be  required  to  multiply  4^2  —  6ab  +  653  by  2a^  — 
Sab  +  4^. 

4flr2  _  6ab  +  6J2 

2^2  -  3ah  4-  4^ 
Sa^  —  lOa^  4-  12«2J2 

—  12a^  +  15a^^  —  18ab^ ' 

+16g2^  — 20^53+  24^  - 

8a^  —  22a^  +  43^2^  —  38ab^  +  24^. 


MULTIPLICATIOIT.  21 

7.  Let  it  be  required  to  multiply  2a^  +  3ic  -f  4  by  2a;2_  3^  _l_  4 
2a;2  4.  3a:  +  4 

4a4  ^  6^  ^  8a;2 

—  6a^  —  9x^^  —  12a; 

+  Sx^  4-  12a;  +  16 

4a4  ^  7a;2  _|_  16. 

MULE. 

Multiply  every  term  of  the  multiplicand  hy  each  term  of  the 
multiplier  in  succession  ;  if  a  term  in  the  multiplicand  and  a 
term  in  the  multiplier  have  like  signs,  prefix  the  sign  +  to 
their  product ;  if  they  have  unlike  signs,  prefix  the  sig7i  —  ; 
then  take  the  sum  of  these  partial  products  to  form  the  complete 
product, 

1.  Multiply  2p^qhj  2q  +  p.  Ans.  dpq  +  2^2  _  2q^, 

2.  Multiply  a^  +  Sab  +  2b^  by  7a  —  55. 

Ans.  W  4-  l^a%  —  aW'  -  IW, 

3.  Multiply  a^  —  ah^-  W  by  a^  +  ah  —  b^ 

Ans.  a^  —  aW  +  2ab^  —  ¥. 

4.  Multiply  a^  —  ab-^  W  by  a^  +  «5  —  W. 

Ans.  a*  —  a^y^  +  ^aW  —  45*. 

5.  Multiply  or2  4-  2ax  4-  x^  by  a^  4-  ^ax  —  x\ 

Ans.  a^  +  ^aH  4-  ^a^x^  —  a:*. 

6.  Multiply  a2  _]_  4^3;  4.  4a;2  by  a^  —  4aa:  +  4a:2 

Ans.  a*  —  8a2a:2  4.  iBrc^. 

7.  Multiply  a2  __  <jiax  -\-  bx  —  x^hj  b  +  x. 

Ans.  a%  +  {a  —  bfx  —  2ax^  —  a^. 

8.  Multiply  15a;2  4.  isax  —  Ua^  by  4:X^  —  2ax  —  a\ 

Ans.  60x^  4-  42a2;3  -  lOlla^x'^  +  lOa^x  +  14a*. 

9.  Multiply  22)3  4.  4arJ  4.  8a;  4-  16  by  3a;  -  6. 

Ans.  6a;4  — 96. 


2a  FUNDAMENTAL    PROCESSES. 

10.  Multiply  7?  —  7^y  ■\-  xyl^  —  if  by  x  ■\-  y. 

Ans.  7^  —  x^y^  +  x^y^  —  if. 

11.  Multiply  a^  -\-h^  -{-  c^  -^Ic  ■}-  ac  —  ab  by  a  +  h  —  c, 

Ans.  a3  +  Z>3  _  ^  ^  ^ahc. 

12.  Multiply  a:*  +  2a:3  ^  ^  _  4a;  __  11  by  a,-2  —  2a:  +  3. 

^ws.  n«  +  10a;  —  33. 

13.  Multiply  a*  —  2(z3  +  3^2  -  2a  +  1    by   a^  +  2a3  +  3^2  -f 
2a  H-  1.  Ans.  a^  +  2a«  +  3a*  +  2a^  +  1. 

14.  Multiply  together  a  —  x,  a  +  Xy  and  a^  +  a:^. 

Ans.  a*  —  x^. 

15.  Multiply  together  x  —  3,  a;  —  1,  a;  +  1,  and  a;  +  3. 

Ans.  x^  —  10a:2  +  9. 

16.  Multiply  together  a^--a;  +  l,  ar^+a;  +  l,  and  a^— a^^  +  l. 

Ans.  a:^  H-  ic*  +  1. 

17.  Multiply  together  a  -}-  x,  b  -^  x,  and  c  +  x. 

Ans.  abc  +  (ab  +  be  -\-ac)x  -\-  (a  -t  b  -{■  c)x^  +  7?. 

18.  Simplify  (a  +  b){b  +  c)  —  {c-\-d)  (a  +  o?)  —  (a  +  c)  (^>— ^). 

19.  Simplify  (a  +  J+  c+  ^)2  +  (a  -^  &  —  c  +  ^)^+  («—  ^  + 
c— (?)2  -f-  (a  +  J  —  c  —  dy.  Ans.  4  (a2  +  ^2  +  ^2  +  d^). 

20.  Simplify  (a  +  J  +  c)^  —  a  (J  +  c  —  a)  —  J  (a  +  c  —  J)  — 
c(a  +  Z,  _  c).  Ans.  ^{a?  +  b"^ -{-  c^). 

21.  Prove  that         a^  j^  f  ^  [x  -^  yf  =  2  (x^  +  xy  -\-  y^f  + 
^7?y'^{x-\-yf{3^-{-xy-\-y^). 

22.  Prove  that  4a;2^  (ar^+  y2)  =  (.r2_|-  xy-\-y^f—  {x^-^xy^y'^f. 

23.  Multiply  (a;2  -  3a;  +  2)2  by  a^^  +  6a;  +  1. 

^?i5.  a;«  —  22a;*  +  60a;3  _  55^  ^  12a;  +  4. 

24.  Multiply  (a  +  «»)2  by  (a  -  Z')^. 

^ws.  aS  —  a*5  —  2a3&2  _^  2a2Z»3  +  aJ4  __  js. 

25.  Prove  that  (a  +  If  -  (a  —  &)2  =  4a&. 


MULTIPLICATIO]^^.  23 

71.  The  square  of  the  sum  of  tico  quantities  is  equal  to  the 
sum  of  their  squares  increased  by  twice  their  product. 

If  we  multiply 

by 


a  -\-b 
a  -\-b 

a^-^ab 
-\-ab-\-l^ 

Tve  obtain  a^  +  2ab  -{■  l^ ',  hence  {a-\-hf=a^-\-2ab-\-J?, 

If  we  wish  to  obtain  the  square  of  the  sum  of  two  quantities, 
this  theorem  enables  us  to  write  the  terms  of  the  result  without 
the  necessity  of  performing  the  actual  multiplication. 

EXAMPLES. 

1.  (2  +  5)2  =  4  +  20  +  25  =  49. 

2.  (2w  +  3/2)2  _  4^2  _|_  i2mn  +  9^2. 

3.  (ax  +  byf  =  a^x^  +  "^abxy  +  Wy\ 

4.  (c  4-  2^)2  =  c2  +  4c<Z  +  4c?2. 

5.  (^2  +  J2)2  ^  ^4  ^  2^252  +  54. 

6.  (a3  +  J3)2  =:  ^6  ^  2a3^>3  ^  ^6. 

7.  [(a;  +  yY  -^{x-  7/)«]2  =1  (2:  +  y?"^  +  2(a;  +  «/)"»(a;  -yY  + 
(a;-2/)2". 

72,  27^6  square  of  the  difference  between  two  quantities  is 
equal  to  the  sum  of  their  squares  diminished  by  twice  their 
product, 

K  we  multiply    a  —b 
by  a  —b 

a2_  ab 
—  ab^-U^ 


we  obtain  a^  —  %ab  +  b^\  hence  {a—bY=^a^—%ab-\-b\ 


EXAMPLES, 


1.  (5  -  3)2  =  25  -  30  +  9  =  4. 

2.  (2a;  —  yf  =  4a;2  —  4a;2/  +  y\ 

3.  (3rc  —  5^)2  =  9ic3  —  30a;i2  + 


24:  FUNDAMENTAL    PROCESSES, 

4.  (c  —  2dy  =  c2  -  4c^  +  4:cP. 

5.  (a2  _  ^2)2  =  ^4  _  2a2^  ^  j*. 

6.  [a  +  b-(c-\-  d)Y  =  a^  +  2aJ  +  ^  -2  (a  +  5)  (c  +  ^)  + 

73.  ^e  product  of  the  sum  and  the  difference  of  two  quan- 
tities is  equal  to  the  difference  between  their  squares. 


K  we 

multiply 

a  -\-b 

by 

a  -b 

a^  +  ab 
-ab- 

-^ 

we 

obtain 

a^- 

-^; 

^;  hence  (a-\-b)  {a—b)=a^—b^. 


EXAMriiJSS. 

1.  (3  +  2)  (3  -  2)  =  9  -  4  =  5. 

2.  (3a  +  2*)  (3a  -  2b)  =  9a^  -  AJP. 

3.  (m  +  1)  (m  — 1)  =m2  — 1. 

4.  (c  +  2d)  (c  —  2d)  =  (^-  4^. 

5.  (a2  4-  ^)  (a2  _  ^>2)  =  ^  _  J4. 

6.  [(a  +  J)  +  c]  [(a  4-  *)  -  c]  =  a^  +  2aJ  +  ^  -  c». 

7.  [(a  +  J)2  +  (a:  -  y)T  [{a  +  ^)2  -  (a:  -  2^)2]  =  (a  +  J)*  - 

7^.  Meaning  of  the  Sign  ±. 

We  may  here  indicate  the  meaning  of  the  double  sign  ±, 
which  is  sometimes  used. 

Since  (a  +  b)^  =  a^  +  2ab  +  V^,  and  (a  -  bf  =  a^  -  2ab  +  l^, 
we  may  write  both  formulae  in  the  following  abbreviated  form : 

(a  ±  bf  =  a^±  2ab  +  2^. 

By  this  notation  we  are  enabled  to  express  two  different 
theorems  by  one  formula.  The  expression  a  ±  &  is  read  a  plus 
or  minus  b. 


MULTIPLICATION.  25 

75.  By  the  aid  of  the  preceding  theorems  the  process  of  mul- 
tiplication may  often  be  abridged.     Thus, 

(aj^-bj^c-^d)  {a-\-b—c—d)^\{a  +  h)-^{c  +  d)]  \{a-^h)  —  {c-\-d)'\ 
=  {a  +  hf-{c  +  df  (73)  =a?+  2ah  +  ^2  _  (^2  ^_  <^cd  +  d^)  (71) 
=a^-\-'itah-\-V^—(?—'^cd—d?. 

REMARKS   ON  MULTIPLICATION. 

76.  The  degree  of  the  product  of  two  monomials  is  equal  to 
the  sum  of  the  degrees  of  the  multiplicand  and  multipher,  since  all 
the  factors  of  both  monomials  appear  in  the  product  (69,  Prin.  2). 
Thus,  if  we  multiply  'ia%  which  is  of  the  third  degree,  by  Zab\ 
which  is  of  the  fourth  degree,  we  obtain  Qa%\  which  is  of  the 
seventh  degree.  Hence,  if  two  polynomials  are  homogeneous, 
their  product  will  be  homogeneous  (70,  6). 

77.  The  number  of  terms  in  the  product  of  two  polynomials, 
when  the  partial  products  do  not  contain  similar  terms,  is  equal 
to  the  product  obtained  by  multiplying  the  number  of  tenns  in 
the  multiplicand  by  the  number  of  terms  in  the  multiplier.  Thus, 
if  there  be  m  terms  in  the  multiplicand,  and  n  terms  in  the  mul- 
tiplier, the  number  of  terms  in  the  product  will  be  mn. 

If  the  partial  products  contain  similar  terms ,  the  number  of 
terms  in  the  product  after  reduction,  will  evidently  be  less  than 
7nn. 

78.  When  the  multiplicand  and  multipher  are  arranged  in 
the  same  way,  according  to  the  powers  of  some  common  letter,  if 
there  be  one,  the  first  and  last  terms  of  the  product  are  unlike 
any  other  terms.  Thus,  in  the  sixth  example  of  Art  70,  the  mul- 
tiplicand and  multiplier  are  arranged  according  to  the  descending 
powers  of  a ;  the  first  t^rm  of  the  product  is  8a*  and  the  last  term 
is  24S*,  and  there  are  no  other  terms  which  are  like  these ;  for 
the  other  terms  contain  a  raised  to  some  power  less  than  the 
fourth,  and  thus  differ  from  8a*;  and  they  all  contain  a  to  some 
power,  and  thus  differ  from  24 J*.  Therefore  the  product  of  two 
polynomials  cannot  contain  less  than  two  terms. 


FUNDAMENTAL    PROCESSES. 


79. 


SYNOPSIS    FOR    REVIEW. 


CHAPTER  11— Continued. 
MULTIPLICATION. 


iMuUiplicand, 
Multiplier. 
Product. 
Factors. 

^  (  Order  of  factors. 

Product  of    ) 

Monomials.     )  ^^,„    -  ^Coefficients. 

{Law  of  .    .    I  Exponents. 

/  Investigation  for  rule. 
Product  op    )  n^u. 


Theorems 


^Remarks. 


Degree  of  product. 
Product,  homogeneous  when. 
Number  of  terms  in  product. 
First  and  last  terms,  different 
•when. 


DIVISION. 


80.  Division  is  the  converse  of  Multiplication.  In  Mul- 
tiplication we  determine  the  product  of  given  factors.  In  Di- 
vision we  have  the  product  of  two  factors,  and  one  of  them  given 
to  determine  the  other  factor. 

81.  Tlie  Dividend  is  the  given  product. 

82.  The  Divisor  is  the  given  factor. 

83.  Tlie  Quotient  is  the  factor  to  he  determined. 


DIVISION.  27 

84.  To  find  the  Quotient  of  two  Monomials. 

.  Let  it  be  required  to  divide  Soa^^c^  by  7a^b^c. 
The  division  may  be  indicated  thus : 

naWc  ' 

Now,  since  the  quotient  must  be  such  a  quantity  that  when 
it  is  multiphed  by  the  divisor  the  product  shall  be  equal  to  the 
dividend,  the  coefficient  of  the  quotient  multiplied  by  7  must 
give  35 ;  hence,  the  coefficient  of  the  quotient  is  found  by 
dividing  35  by  7.  Again,  the  exponent  of  any  letter  in  the 
quotient  added  to  the  exponent  of  the  same  letter  in  the  divisor, 
must  give  the  exponent  of  this  letter  in  the  dividend  (69,  Prik.  2) ; 
hence,  the  exponent  of  any  letter  in  the  quotient  is  found  by 
subtracting  its  exponent  in  the  divisor  from  that  in  the  dividend. 
Therefore, 

-r-^v„—  =  5a^^c.     Hence, 

Prin"CIPLES. — 1.  Tlie  coefficient  of  the  quotient  of  two  given 
monomials  is  the  quotient  obtained  by  dividing  the  coefficient  of 
the  dividend  by  that  of  the  divisor. 

2.  Every  letter  which  occurs  in  the  dividend  must  be  tvritten 
in  the  quotient,  ivith  an  exponent  lohich  is  found  by  subtracting 
its  exponent  in  the  divisor  from  that  in  the  dividend. 

Cor.  —  =  «'"-'«  =  fl",  and  —  =  1 : 

hence  a"  =  1. 


1.  Divide  abx  by  x.  Ans.  ab. 

2.  Divide  Qa^xy  by  Zax,  Ans.  2ay. 

3.  Divide  12abc^mn  by  Sbchi.  Ans.  ^anu 

4.  Divide  Iha^T^  by  ^a^x\  Ans.  haH. 

5.  Divide  27a"»  +  "a:^  +  "  by  9a"a^.  Ans.  Sa^'ai^. 


28  FUNDAMENTAL    PROCESSES. 

85.  It  follows  from  Art.  84  that  the  exact  division  of  mono- 
mials will  be  impossible : 

1st.  When  the  coeflBcient  of  the  dividend  is  not  divisible  by 
that  of  the  divisor. 

2d.  When  the  exponent  of  a  letter  in  the  divisor  is  greater 
than  the  exponent  of  the  same  letter  in  the  dividend. 

3d.  When  the  divisor  contains  a  letter  that  is  not  found  in  the 
dividend. 

86.  To  find  the  Quotient  of  two  Polynomials. 
1.  Let  it  be  required  to  divide  ab  —  he  by  1). 

— 7 —  =  a  —  c ;  for  {a  —  c)h  =  ah  —  be. 

In  this  example,  we  observe  that  corresponding  to  the  term 
ah  in  the  dividend  and  to  the  divisor  h,  there  is  the  term  a  in 
the  quotient ;  and  corresponding  to  the  term  —  he  in  the  div- 
idend and  to  the  divisor  h,  there  is  the  term  —  c  in  the  quo- 
tient. 

We  have  already  seen  that 

ix(-c)  =  -  he,  and  (-  h)  (-  e)  =  he  (70). 

In  like  manner,  the  following  statements  may  be  admitted : 

—  he      ^        :i    he  , 

=  hf  and  =z  —  h. 


—  c  —c 

Thus  the  sign  of  the  quotient  is  deduced  from  the  signs  of  the 
dividend  and  divisor  by  the  rule. 

Like  signs  produee  +,  and  unlike  signs  produee  — . 

2.  Let  it  be  required  to  divide  ah'^  —  ahe  -\-  ahd  by  ah. 
at^  —  ahe  +  ahd 


ah 


—  l^c-^d. 


We  divide  each  term  of  the  dividend  by  the  divisor,  then 
collect  the  partial  quotients  to  obtain  the  complete  quotient. 


Divisio]^-.  29 

3.  Let  it  be  required  to  divide  8a^  +  43«2^»2  _  22a^  +  245^— 
S8ab^  by  2a^  +  U^  —  3a6. 

The  operation  may  be  conveniently  arranged  as  follows : 


DIVIDEND. 

8a^—22a^  +  ^3a^b^—dSa¥  +  24.¥ 
8a^—12a^-\-16a^^ 


DIVISOR. 

2a2_3«&4-4Z>2 


4a2_5a<$>  +  (Jj2--Quo. 


1st  Rem.  =  -10a^-\-27aW—38ab^-\-2i¥ 

—lOa^  +  15a^b^—20ab^ 
2dEem.  =  12aW—18aI)^  +  2U^ 

12aW—18ab^+24:¥ 

Now,  the  term  of  the  dividend,  which  contains  the  highest 
power  of  any  letter  as  «,  must  be  equal  to  the  product  arising 
from  multiplying  the  term  of  the  divisor  which  contains  the 
highest  power  of  that  letter  by  the  term  of  the  quotient  whicli 
contains  the  highest  power  of  the  same  letter.  Therefore,  if  we 
arrange  the  dividend  and  divisor  according  to  the  descending 
powers  of  a  common  letter  as  a,  the  first  term  of  the  quotient  js 
found  by  dividing  the  first  term  of  the  dividend  by  the  first  term 
of  the  divisor.  Hence,  in  this  example  the  first  term  of  the  quo- 
tient is  tt-t;  =  4^2. 
2a^ 

Again,  the  dividend  is  equal  to  the  sum  of  the  partial  pro- 
ducts obtained  by  multiplying  the  divisor  by  each  term  of  the 
quotient  in  succession  ;  and,  therefore,  if  the  product  of  the  divi- 
sor by  the  term  just  found  is  subtracted  from  the  dividend,  the 
remainder  must  be  equal  to  the  sum  of  the  partial  products  ob- 
tained by  multiplying  the  divisor  by  the  remaining  terms  of  the 
quotient,  and  hence  may  be  used  as  a  new  dividend  to  obtain  the 
second  term  of  the  quotient.  Proceeding  in  this  manner,  we  find 
the  complete  quotieut  to  be 

4^2  —  5ab  +  6^. 

A  similar  course  of  reasoning  is  applicable  when  the  dividend 
and  divisor  are  arranged  according  to  the  ascending  powers  of  a 
common  letter. 


30  FUNDAMENTAL    PROCESSES. 

JR  VLB. 

I.  Arrange  the  dividend  and  divisor  according  to  the  poivers 
of  some  common  letter. 

II.  Divide  the  first  term  of  the  dividend  by  the  first  term  of 
the  divisor ;  the  result  will  be  the  first  term  of  the  quotient  Miil- 
tiply  the  whole  divisor  by  this  term,  and  subtract  the  product 
from  the  dividend. 

III.  Regard  the  remainder  as  a  neiu  dividend  ;  find  the  second 
term  of  the  quotient  in  the  same  manner,  and  proceed  with  it  as 
with  t1i£  first  term  ;  and  so  on. 

Remarks. — 1,  Tlie  situation  of  the  divisor  in  regard  to  the  dividend  is 
a  matter  of  arbitrary  arrangement ;  but  by  placing  it  on  the  right  it  is  more 
easily  multiplied  by  the  several  terms  of  the  quotient  as  they  are  found. 

2.  Wlien  there  are  more  than  two  terms  in  the  quotient,  it  is  not  neces- 
sary to  bring  down  any  more  terms  of  the  remainder,  at  each  successive 
subtraction,  than  are  required  by  the  quantity  to  be  subtracted. 

3.  It  is  evident  that  the  exact  division  of  one  polynomial  by  another  will 
be  impossible,  when  the  first  term  of  the  arranged  dividend  is  not  exactly 
divisible  by  the  first  term  of  the  arranged  divisor  ;  when  the  last  term  of 
arranged  dividend  is  not  divisible  by  the  last  term  of  the  arranged  divisor, 
or  when  the  first  term  of  any  arranged  remainder  is  not  divisible  by  the  first 
term  of  the  divisor. 

EXAMPLES. 

1.  Divide  a:^  +  1  by  re  +  1.  Ans.  x^  —  x  -]-l, 

2.  Divide  213^  +  Sy^  by  3x -^  2y^      Ans.  Qx^  —  6xy  +  AyK 

3.  Divide  a^  —  2a^  -{- 1)^  hj  a  —  b.     Ans.  a^  -\-  ab  —  b^. 

4.  Divide  a^  —  2a^b  —  ?>al^  by  a  +  ^>.  Ans.  a^  —  Zab. 

5.  Divide  64^  —  \f  by  2x  —  y. 

Ans.  32a:5  _^  16^4^  ^  %^y%  4.  4a4j^  _^  '^xy'^  +  if. 

6.  Divide  a^  -\-lr'  by  «  +  b. 

Ans.  a*  —  a%  -{■  aW  —  a¥  +  b\ 

K.  Divide  a«  —  IQa^x^  +  64:^:6  by  4^2  4.  ^^2  _  4^ax. 

Ans.  16x^  +  IQas^  +  12a^x^  +  4:a^x  +  aK 


Divisio:^:.  31 

8.  Divide  1  —  ISz^  +  Sl;^*  hj  1  +  6z  +  9z^. 

Ans,  1  —  62;  +  9zK 

9.  Divide  81a^  +  16^12  _  72a4J6  by  9^^  +  12^2^,3  ^  4J6. 

^?zs.  9^4  _  I2a2^,3  _j_  4ja, 

10.  Divide  a^  —  x^y  +  a^y^  —  a:2^^  +  ^y^  —  y^  by  0^  —  y\ 

Ans.  x^  —  xy  ■\-  y\ 

11.  Divide  x"^  +  a?  —  ^x^ -\- hx  —  ^  hj  x^  +  'Zx  —  3. 

Ans.  ic2  —  a;  +  1' 

12.  Divide  a*  +  2^252  _j_  9^4  by  ^2  _{_  ^ab  +  3^»2. 

^^5.  ^2  _  2ah  +  3^>2. 

13.  Divide  a^  —  ¥  by  a^  +  2a2Z>  +  2«Z^2  ^  js, 

J?z5.  a3  _  2^25  ^  2a52  _  J3. 

14.  Divide  7^  —  27?-\-l  by  a^  —  22;  +  1. 

Ans.  a:*  +  2.r3  +  3a:2  _|.  2a;  +  1. 

15.  Divide  a^  +  a^h  -\-  a^c  —  ahc  —  hH  —  hc^  by  a^  —  Ic. 

Ans.  a  -\-h  +  c. 

16.  Divide  a^  +  1^  —  (^  +  dale  hj  a  +  h  —  c. 

Ans.  a^  -\-b^  +  d^  -\-ac-\-hc  —  ab. 

17.  Divide  1  —  Qa:^  —  Sa:^  by  1  +  2a;  +  x\ 

Ans.  1  —  22;  +  3a;2  —  4a:3  ^  5^4  _.  ca^s  ^  7^^  _  ga;^ 

18.  Divide  {a  +  b  —  c)  {a —  b  +  c)  ib  -\- c  —  a)  hy  a^  —  b^ — 
c^  H-  2bc.  Ans.  b  -\-c  —  a. 

19.  Divide  {a^  —  bcf  +  8^c8  by  a^  +  ^<?. 

^w5.  a^  —  U^c  4-  7*2^2. 

20.  Divide  the  product  of  a;^  _  2a;  +  1  and  a:^  _  3^;  +  2  by 
ic3  _  32-2  _|.  3a;  _  1.  ^1^25.  ic3  _^  3^2  ^  a;  _  2. 

21.  Divide  the  product  of  a^  -\- ax  -\-  7?  and  a^  +  x^  hy  a^  -\- 
a^T?  +  7^.  Ans.  a  +  X. 

22.  Divide  a^  {b  -\-  c)  —  b^  {a  +  c)  -^  c^  {a -\- b)  +  abc  by  «  — 
b  +  c.  Ans.  ab  -\-  be  +  ac. 

23.  Divide  aa;2  —  aV^  +  ^'2a;  _  x^  by  {x  -\-b)  {a  —  x). 

Ans.  X  —  b. 

24.  Show  that   (a;2  —  a:?/  +  y'^f  +  {x^  +  xy  -\-y^y  is  divisible 
by  2a;2  +  2y\ 


32  FUKDAMENTAL    PROCESSES. 

87.  Divisibility  of  Quantities  of  the  form  of  x^^  ±  ap. 

1.  of  —  a""  is  divisible  hy  x  —  a  when  w  is  a  positive  integer. 

2.  a:"  —  a"  is  divisible  by  a;  +  a  when  w  is  an  even  positive 
integer. 

3.  a;"  +  a"  is  divisible  hj  x  -\-  a  when  n  is  an  odd  positive 
integer.* 

:exa  MPLSes. 

1.  Divide  x^  —  a^  by  a;  +  a.  Ajis.  x  —  a. 

2.  Divide  a;^  +  a^  by  a;  -f  a.  Ans.  7?  —  ax  -\-  a^. 

3.  Divide  x^  —  a^  by  ar  4-  «.  Ans.  x^  —  ax^  ■\-  ahc  ^  a\ 

4.  Divide  7^  -\-  a^  hj  x  -\-  a. 

Ans.  x^  —  a7?  +  ah^  —  ah:  -\-  a*. 

5.  Divide  a^  —  b^  by  a  —  h. 

Ans.  a*  +  a^  +  a-b^  +  ab^  +  b*. 

6.  Divide  a^  —  b^  hj  a  +  b. 

Ans.  a^  —  a*b  +  a^b^  —  a^^^  ^  aV^  —  lP. 

7.  Divide  a?»  +  1  by  a;  +  1-  Ans.  x^  —  x  -\-  1. 

8.  Divide  a:^  _^  i  by  a;  +  1.       Ans.  a^  —  a^-{-x^  —  x-^1. 

9.  Divide  a^  —  1  by  a:  +  1.  ^ws.  x  —  1. 

10.  Divide  a;*  —  1  by  a;  +  1.  ^W5.  a^  —  a:^  +  a;  —  1. 

11.  Divide  a:^  —  1  by  a;  +  1. 

Ans.  x^  —  x^-{-a^  —  x^  +  x  —  l. 

12.  Divide  a^  —  1  by  a:  —  1.         Ans.  a;  +  1. 

13.  Divide  a;^  —  1  by  a;  —  1.         Ans.  a;^  -f  a;  +  1. 

14.  Divide  a:r*  —  1  by  a;  —  1.         Ans.  a^  -{■  x/^  -\-  x  +  1. 

15.  Divide  a;^  —  1  by  a:  —  1.         Ans.  xf^-\-a^-\-a^-\-x-^l. 

The  student  should  carefully  observe  the  law  of  the  signs  and  expa 
nents  in  the  preceding  examples. 

*  In  Chapter  XVII  we  shall  give  a  general  proof  of  these  statements.    It 
will  be  easy  for  the  student  to  verify  them  in  any  particular  case. 


FACTORING.  83 


FACTORING. 

88.  Factoring  is  the  process  of  resolving  a  quantity  into 
its  factors. 

89.  A  Prhne  Quantity  is  one  which  is  exactly  divisible 
only  by  itself  and  by  1.  Thus,  x,  y,  and  a  -{-  h  are  prime  quanti- 
ties; but  xy  and  ax  +  az  are  not  prime. 

90.  Two  quantities  are  said  to  be  prime  to  each  other,  or 
relatively  prime,  when  they  have  no  common  factor.  Thus,  ab 
and  ccl  are  relatively  prime. 

The  unit  1  is  not  generally  considered  as  a  factor. 

91.  A  Composite  Quantity  is  one  which  is  the  pro- 
duct of  two  or  more  factors.  Thus,  a^  —  h^i^o,  composite  quan- 
tity, the  factors  of  which  are  a  -\-  h  and  a  —  l. 

93.  To  resolve  a  monomial  into  its  prime  factors. 

RULE. 

To  the  prime  factor fi  of  the  numerical  coefficient  annex  the 
prime  factors  of  the  literal  part, 

EXAMPLES. 

1.  Eesolve  VZa^  into  its  prime  factors.      Ans.  2  x  2  x  ^aat, 

2.  Resolve  IScrJ^  into  its  prime  factors.      Ans.  2  x  3  x  ^ahK 

3.  Resolve  21m^n^x  into  its  prime  factors. 

Ans.  7  X  Smmmnnx, 

4.  Resolve  ^^a^bx^y^  into  its  prime  factors. 

Ans.  7  X  laabxxyyy, 

5.  Resolve  ^lOax^yz^  into  its  prime  factors. 

Ans.  2  X  3  X  5  X  laxxxyzz. 

6.  Resolve  %^m^x^yz  into  its  prime  factors. 

Ans.  2  X  ISmmmmxxyz, 


84  FUNDAMENTAL     PROCESSES. 

93.  To  resolve  a  polynomial  into  two  factors,  one 
of  which  shall  be  a  monomial. 

RULE. 

Divide  the  given  quantity  hy  any  monomial  that  will  exactly 
divide  each  of  its  terms ;  the  divisor  vnll  be  o?ie  factor,  and  the 
quotient  the  otiier. 

JEXAMPZES. 

Resolve  each  of  the  following  expressions  into  two  factors, 
one  of  which  shall  be  a  monomial : 

1.  a  -\-  ax.  Ans,  « (1  +  x). 

2.  xz  -!-■  yz.  Ans.  z{x  -\-  y). 

3.  ic^y  _j_  xy^.  Ans.  xy  {x  +  y). 

4.  6a^  +  ^a^c.  Ans.  Sab  {2b  +  3ac). 

6.  25a*  —  30a^  +  15^2^.  Ans.  5a^  (5a^  —  Gab  +  362). 

6.-  24a2^c.i-  —  SOa^b^c^y  +  SGaWcd  +  Qabc. 

Ans.  6abc  {4abx  —  ba'^b^c^y  -f  Ga^b'^d  +  1). 

7.  3a2a;  +  6abx  +  3b^x.  Ans.  Sx  (a^  -\-  2ab  +  b^). 

8.  5  —  5y.  Ans.  5  (1  —  y). 

9.  42a2J2  _  <^abcd  +  7abd.  Ans.  '7ab  (6ab  —  cd  +  d). 

10.  aly^c  +  5aJ3  4.  ^52^2.  Ans.  aW-  (c  +  52>  +  C^). 

11.  ex  —  Zcxz  -\-  cx^.  Ans.  ex  (1  —  3^  +  a;). 

12.  Vl(^b7?  —  \h(?7^  —  G(?:i?y.       Ans.  Zc^t?  {4:C-b  —  5cx  —  2y). 

In  resolving  a  polynomial  into  two  factors,  one  of  which  shall 
be  a  monomial,  it  is  common  to  divide  by  the  greatest  monomial 
that  will  exactly  divide  each  of  its  terms ;  but  it  is  not  necessary 
to  do  this.  Thus,  x^y  +  xy^  may  be  expressed  under  any  one  of 
the  three  following  forms : 

^y  (^  +  y)y      ^  {^y  +  y%      y  {^^  +  xy). 


FACTOEIKQ.  35 


PRINCIPLES  USED  IN  FACTORING  BINOMIALS. 

94.  The  difference  letween  the  squares  of  two  quantities  is 
equal  to  the  product  of  the  sum  and  the  difference  of  the  quanti- 
ties (73).     Thus, 

^2  _  J2  ^  (^  +  ^)  (^  _  ^). 

95.  The  difference  iettueeti  any  tiuo  like  poivers  of  two  quan- 
tities is  divisible  hj  the  difference  between  the  quantities  (87). 
Thus, 

3  73 

^  ~     '  =  a^-[-ab  +  b'^)  whence,  a^—  b^=  {a  —  b)  {a^  +  ab  +  b^). 
a  —  0 

96.  The  diff^erence  between  any  two  like  even  powers  of  tivo 
quantities  is  divisible  by  the  sum  of  the  quantities  (87).     Thus, 

^4  _  J4 

i-  z=z  a^—  a%  +  ab^  —  b^ ;  whence,  «*—  J*  = 

a  -\-b 

{a  +  b){a^  —  a%-\-a¥—b% 

97.  The  sum  of  any  tiuo  like  odd  powers  of  tiuo  quantities  is 
divisible  by  the  sum  of  the  quantities  (87).    Thus, 

^1±-?.  =  a2_  ab  +  b^',  whence,  a^  +  b^=  {a  +  b)  (a^-  ab  +  b^), 
a  -J-  u 

EXAMTLES. 

98.  Eesolve  each  of  the  following  expressions  into  its  prime 
factors : 

Ans.  {a  -\-  c){a  —  c). 

Ans.  (2x  +  y)  {2x  —  y). 

Ans.  (z  -{'l)(z^  —  z-\-  1). 

Ans.  {a^-\-b^){a  +  b)(a-b), 

Ans.  {x  +  y)(x^  —  xhj  +  xhf  —  xy^  +  y% 

Ans.  (a^  +  (^)  {a^  +  ^2)  («  +  c)  {a  —  c). 

Ans.  {x!^  +  y^)  (^^  +  y)  {^^  —  y\ 

Ans.  (1  +  (^){l  +  c)  (1  -  c). 
Ans.  (Sx  +  1)  (dxi  -dx-i- 1). 
Ans.  l2x  -  1)  (4a;2  -\- 2x  i- 1). 


1. 

a^- 

-c2. 

2. 

^x^- 

-f. 

3. 

z^-[-  1. 

4. 

a^- 

-¥. 

5. 

o^  +  yK 

6. 

a^- 

-C8. 

7. 

a^- 

-t' 

8. 

1- 

■  (^. 

9. 

27a^»  +  1. 

:o. 

8x3 

-1. 

36  FUNDAMENTAL     PEOCESSES. 

99,  Certain  trinomials  can  be  factored  in  accordance  with 
the  following  principle : 

If  two  terms  of  a  trinomial  are  positive  squares,  and  the  other 
term  is  twice  the  product  of  the  square  roots  of  these  two,  the 
trinomial  is  equal  to  the  square  of  the  sum,  or  the  square  of  the 
DIFFERENCE,  of  these  squarc  roots,  according  as  that  other  term 
is  positive  or  negative  (71-73).    Thus, 

a^  ±  2ab  -{^  =  (a  ±  hf  z=(a±h)ia±  h). 

EXAMPLES. 

Resolve  each  of  the  following  ten  expressions  into  two  equal 
fectora: 

1.  ir»  -f  2fla;  +  a\  Ans.  {x  ■\-a){x-\-  a), 

2.  7n^  -f  w*  +  2^2/^2.  Ans.  (m^  +  n^)  (m^  +  n^)- 

3.  16a^b^m^  —  Sa^hhn  +  1.       Ans.  (^a^U^m.  —  1)  (4:a^li^m  —  1). 

4.  36a2  -f  12ab  +  R  Ans.  (6a  +  b)  (6a  +  b). 

5.  c2  —  lOcd  +  25^.  A71S.  {c  —  5d)  (c  —  5d). 

6.  ah^  +  2aa^y  +  f.  Ans.  (ax^  •+  y)  {ax^  +  y), 

7.  25x^y*  +  20x3^2^  +  4cZ^.         Ans.  (5xy^  +  2z)  {5xy^  +  2z). 

8.  Ore*  —  63^z^  +  ^.  ^ns.  (3a^—  z^)  (Sx^  —  z^), 

9.  (a  +  J)3_  2(a  +  b){c  +  d) -{- (c -\-  d)\ 

Ans.  [a-^b  —  {c-\-d)'][a  +  b  —  (c-\-  d)\ 

10.  a^*"  4-  2a'»5"  +  ll^\  Ans.  («"*  +  6»)  (a"*  +  ^>"). 

11.  Can  x^  —  2xy  —  y^  be  resolved  into  two  equal  factors? 

12.  Resolve  ^y^(?  —  (^  4-  c^  —  fl2)2  into  its  prime  factors. 
Here  we  have  the  difference  between  two  squares;  hence, 

4J2c2  _  (&2  +c2  —  a2)2  =  (2Jc  4.  J2  _^  ^2  _  ^2)  (2^^  _  ^>2  _  ^2  4.  «2), 

But, 

2Jc+J2+c2— a2=(5  +  c)2— a2=(J4-c  +  a)(5  +  c— a),  and 

25c  —  52  _  ^  _^  ^2  =  «2  _  (J2  _  2JC  +  C2)  =  a2  _  ( J  _  c)2  =: 

(a  +  5  —  c)  (a  —  5  +  c). 
Therefore, 
452^_(^4c2— a2)2=(54-c  +  a)(5  +  c— a)(a  +  5— c)(fl5— 5  +  c). 


FACTOKING.  37 

13.  Resolve  m^  +  ^mn  -^-n^ —  a^  -\-%ab —  W  into  its  prime 
factors. 

This  expression  may  be  put  under  the  form, 

But 

m^  +  2mn  -]-n^=i{m-\-  n)%     and     a^—2ab  +  l^={a  —  b)^; 
hence, 

m^  +  2mn  +n^  —  a^  +  2ab  —  b^  =  {m -{-  nf  —  {a  —  bf  = 
{m  -\-  n  -\-  a  —  b)(m  +  n  —  a  -}-  b). 

100.  The  following  formulae  may  be  yerified  by  performing 
the  operations  indicated  in  their  second  members : 

x^-\-  {a-\-b)x  +  ab  =  {x-}-  a){x-\-b)   .  .  .  (1), 

a^—{a  +  b)x-{-ab=(x  —  a){x  —  b).  .  .  (2), 

a^-\-  (a  —  b)x  —  ab  =  {x  +  a)(x  —  b)   ,  .  .  (3), 

xi—(a  —  b)x  —  ab  =  {x  —  a)(x-\-b)   .  .  .  (4). 

From  (1)  and  (2)  it  follows  that 

Any  trinomial  of  the  form  ofx^  +  mx  +  n,  or  of  the  form  of 
xi  —  mx  -\-  n,  can  be  resolved  into  two  binomial  factors,  if  the 
coefficient  of  the  second  term  is  equal  to  the  sum  of  two  quafitities 
whose  product  is  equal  to  the  third  term. 

From  (3)  and  (4)  it  follows  that 

Any  trinomial  of  the  form  of  x^  +  mx  —  n,  or  of  the  form  of 
x^  —  mx  —  n,  can  be  resolved  into  two  binomial  factors,  if  the 
coefficient  of  the  second  term  is  equal  to  the  difference  of  two 
quantities  whose  product  is  equal  to  the  third  term. 

It  will  be  observed  that  we  have  used  the  words  sum  and 
difference  in  their  arithmetical  sense. 

In  the  first  form  both  of  the  terms  in  each  binomial  factor  are 
positive. 

In  the  second  form  the  second  term  of  each  of  the  binomial 
factors  is  negative. 


38  FUNDAMENTAL     PROCESSEft. 

In  the  third  form  the  second  terms  of  the  binomial  factors 
have  contrary  signs,  the  larger  being  positive. 

In  the  fourth  form  the  second  terms  of  the  binomial  factors 
have  contrary  signs,  the  larger  being  negative. 

EXJ.M1*LES. 

1.  Eesolve  ar^  +  5rr  +  6  into  two  binomial  factors. 

This  comes  under  the  first  form.  Let  us  now  seek  two  num- 
bers whose  sum  is  5  and  product  6.  We  see  that  these  numbers 
are  2  and  3 ;  hence, 

a^  -\-  ox  -h  6  =  {x  -\-  2)  (x  -\-  3). 

2.  Resolve  x^  —  9x  -\-  20  into  two  binomial  factors. 

This  comes  under  the  second  form;  and,  therefore,  since 
4  +  5  =  9,  and  4  x  5  =  20,  we  have 

a^  —  9x  +  20z=z(x  —  4:){x  —  5). 

3.  Resolve  a:^  +  4a;  —  32  into  two  binomial  factors. 

This  comes  under  the  third  form;  ond,  therefore,  since 
8  —  4  =  4,  and  8  X  4  =  32,  we  have 

a5  +  4a;  —  32  =  (a;  +  8)  (a;  —  4). 

4.  Resolve  x^  —  ox  —  6Q  into  two  binomial  factors. 

This  comes  under  the  fourth  form;  and,  therefore,  since 
11  —  G  =  5,  and  11  x  6  =  66,  we  have 

a^  —  5x  —  66  =  {x-{-6)(x  —  11). 

Resolve  each  of  the  following  ten  expressions  into  two  bino- 
mial factors : 

5.  ic2  +  8a;  +  15.  Ans.  {x  +  3)  {x  +  5). 

6.  a;2  +  8a;  4-  7.  Ans.  (x  +  1)  (x  +  7). 

7.  a:2  _  a:  _  6.  Ans.  (x  +  2)  (a;  -  3). 
S.  a^ -{- Sx -\- 2.  Ans.  (x  +  2)  (a;  +  1). 
Q,  a^  —  x  —  72.  Ans.  (x  +  8)  {x  —  9). 

10.  a:2  _  13a;  +  42.  Ans.  (x  -  7)  {x  -  6). 

11.  x^-x  —  42.  A?is.  {x  —  7)  {x  +  6). 


FACTORING. 

12.  a^-x-2. 

13.  x^  +  2^;  -  35. 

14.  x^  —  x  —  30. 

39 

Ans.  (x  4-  1)  {x  —  2). 
^7^5.  {x  —  5)(x  -{-  7). 
A71S.  {x  -{■  6)  {x  —  6). 

101.  Since  a^p-{-{a-\-b)xP-{-ab=(xP-{-a){xP-{-b)     .     .     (1), 

x^P—{a-\-b)xP-{'ab  =  {xP—a){xP  —  b)     .     .     (2), 
x^p+{a—b)xP-ab={xP-{-a){xP—b)     .     .     (3), 
and  a^P—{a  —  b)xP—ab={xP—a){xP-\-b)     .     .     (4), 

it  follows  that  such  expressions  as  a;^  -f-  8x^  +  15,  x^  —  Idx^  +42, 
a^  -{-Sx^  -{-  2,  and  a;^  —  5a;^  —  66  may  be  resolved  into  binojnial 
factors  in  the  same  manner  as  the  examples  of  the  preceding 
Article.     Thus, 

a4_|_8a^  +  15  =  (a:2+3)(ic2+5),        afi-133^-{-4:2=(a^-'7){a>^—6), 

.-^8  +  30:*+  2  =  (x^-\-2)(x^-{-l),        x^—6a^—66  =  (x^^e)(x^—ll). 

MISCELIjANEOUS    examtt^es, 

102,  Kesolve  each  of  the  following  expressions  into  its  prime 
factors : 

1.  0!^  —  X.  A71S.  {x  —  l){x-\-  1)  X. 

2.  3ax^  +  6axy  +  3ai/^.  Ans.  3a  (x  -{-  y)  (x  -\-  y). 

3.  2cx^  —  12cx  +  18c.  Ans.  2c  (x  —  3)  (x  —  3). 

4.  27a  —  ISax  +  3ax^.  Ans.  3a  (3  —  x)  (3  —  x). 

5.  3m^n  —  3mn^.  Ans.  3mn{m  -\-  n)  (m  —  7i), 

6.  2sfi  -{-6x  —  8.  Ans.  2  {x  +  4)  (a;  —  1). 

7.  2a^  +  ^xi  —  70a;.  A7is.  2x  {x  +  7)  (a;  —  5). 

8.  a^  —  W  —  c^  —  %bc.  Ans.  (a^b-^c)  (a—b—c). 

9.  ac  -{-  ad  +  bd  -{-  be. 

Ans.  a(c  -^  d)  -\-  b {c  -j-  d)  =  (a  +  b)  (c  +  d). 

10.  am  +  2bx  +  2ax  +  bm. 

Ans.  a  {m  +  2^;)  -\-  b  (m  -{- 2x)  =  (a  +  b)  (m  +  2x), 

11.  a^  —  ab^.  Ans.  a{a  -\-  b){a  —  b). 

12.  7x^—12x-ir^. 

Ans.  X  {7x  —  b)  —  {7x  —  6)  =  {x  —  l)  (7x  -  5). 

13.  x^  —  x^  —  2x.  Ans.  x{x-{-l){x--  2). 

14.  7^  —  \W  +  9. 

A71S.  {x^  -  9)  (x^  -l)=.{x  +  3)(x-  3)  {x  +  1)  (^  -  1). 

15.  x^  —  17ji?  +  16.  Ans.  {x  +  4)  (a;  —  4)  \x  +  1)  {x  —  1). 


40 


FDIfDAMENTAL     PROCESSES. 


103.  SYNOPSIS    FOR    REVIEW. 

RELATION  TO  MULTIPLICATION. 

TERMS  USED  .    .    . 


Dividend. 

Divisor. 

Quotient. 


M0N0M.-5-M0N0M. 


Law  of  coefficients. 
Law  of  exponents. 

{Like. 


Law  of  signs  .    . 


(  Unlike. 


POLYNOM.  ^POLTNOM. 


TERMS  USED 


1.  Coefficient. 
When  impossible.  ^  2,  Exponent. 

3.  Literal  paH 

i  Investigation  for  rule. 
Rule. 
Proof. 
When  impossiblb. 

r  Prime  quantity. 
J  Relatively  prime. 
j  Composite  quantity. 
I  Prime  factor. 


Eh 
O 
<1 


MONOMIALS— Rule. 

POLTNOML^  WITH  MONOMLiL  FACTOR— RULB. 

BINOMIALS— Principles. 


TRINOMIALS   . 


First  form. 
Second  form. 
Third  form. 
Fourth  form. 


CHAPTER  III. 
POSITIVE  AND   NEGATIVE   QUANTITIES. 


104.  In  Algebra  we  are  sometimes  led  to  a  subtraction 
which  cannot  be  performed,  because  the  subtrahend  is  greater 
than  the'^minuend.     In  the  equation 

a  —  {h  -\-  c)=a  —  l)  —  c, 

it  is  implied  that  J  +  c  is  less  than  a ;  but  suppose  that  «  =  7, 
J  =  7,  and  c  =  3 ;  we  shall  then  have 

7- 10  =  7-7-3  =  - 3. 

In  writing  this  equation,  we  may  be  understood  to  make  the 
following  statement :  It  is  impossible  to  take  10  from  7  ;  hut  if 
7  he  taken  from  10,  the  remainder  will  he  3. 

105.  It  might  at  first  seem  unlikely  that  such  an  expression 
as  7  —  10  should  occur  in  practice ;  or  that  if  it  did  occur,  it 
would  only  arise  either  from  a  mistake  which  could  be  instantly 
corrected,  or  from  an  operation  being  proposed  which  it  was  obvi- 
ously impossible  to  perform,  and  which  must  therefore  be  aban- 
doned. As  we  proceed,  we  shall  find,  however,  that  such  expres- 
sions occur  frequently.  It  might  happen  that  a  —  h  appeared  at 
the  beginning  of  a  long  investigation,  and  that  it  was  not  easy  to 
decide,  at  once,  whether  a  were  greater  or  less  than  h.  The  object 
of  this  chapter  is  to  show  that  in  such  a  case  we  may  proceed  on 
the  hypothesis  that  a  is  greater  than  h,  and  that  if  it  should 
finally  appear  that  a  is  less  than  h,  we  shall  still  be  able  to  make 
use  of  our  investigation. 


42  POSITIVE     AND     NEGATIVE     QUANTITIES. 

106.  Suppose  a  merchant  to  gain  in  one  year  a  certain  num- 
ber of  dollars,  and  to  lose  a  certain  number  of  dollars  in  the  fol- 
lowing year;  .what  change  has  taken  place  in  his  capital? 

Let  a  denote  the  number  of  dollars  gained  in  the  first  year, 
and  J)  the  number  of  dollai'S  lost  in  the  second  year.  Then  if  a  is 
greater  than  h,  the  capital  has  been  increased  hj  a  —  b  dollars. 
If  b  is  greater  than  a,  the  capital  has  been  diminished  by  ^  —  a 
dollars.  In  this  latter  case  a  —  Z>  is  the  indication  of  what  would 
be  pronounced,  in  Arithmetic,  to  be  an  impossible  subtraction ; 
but,  in  Algel)rii,  it  is  found  convenient  to  indicate  the  change  in 
the  capital  by  a  —  b,  whether  a  is  greater  or  less  than  b,  which 
we  may  do  by  means  of  an  appropriate  system  of  interpretation. 
Thus,  if  «  =  $400  and  b  =  ^500,  the  merchant's  capital  has  suf- 
fered a  diminution  of  $100.  The  algebraist  indicates  this  in  sym- 
bols thus : 

400  —  500  =  —  100 ; 

and  he  may  convert  his  symbols  into  words  by  saying  that  the 
capital  has  been  increased  by  —  $100.  This  language  is  far  re- 
moved from  that  of  ordinary  life ;  but  if  the  algebraist  under- 
stands it  and  uses  it  consistently,  his  deductions  will  be  sound. 

107.  There  are  numerous  instances  in  which  it  is  convenient 
to  be  able  to  represent,  not  only  the  magnitude,  but  also  what 
may  be  called  the  quality  of  the  things  about  which  we  may  be 
reasoning.  In  business  transactions  a  sum  of  money  may  be 
gained  or  it  may  be  lost ;  in  a  question  of  chronology  we  may 
have  to  distinguish  a  date  before  a  given  epoch  from  a  date  after 
that  epoch ;  in  a  question  of  position  we  may  have  to  distinguish 
a  distance  measured  to  the  north  of  a  certain  point  from  a  dis- 
tance measured  to  the  south  of  it ;  and  so  on.  These  pairs  of  re- 
lated magnitudes  the  algebraist  distinguishes  by  means  of  the 
signs  +  and  — .  Thus,  if  the  things  to  be  distinguished  are 
gain  and  loss,  he  may  denote  by  -f  a  a  gain  of  a  dollars,  and  then 
he  will  denote  by  —  «  a  loss  of  the  same  extent. 

108.  In  Arithmetic  we  consider  only  the  numbers  repre- 
sented by  the  symbols  1,  2,  3,  4,  etc.,  and  intermediate  fractions. 


POSITIVE     AND     NEGATIVE     QUANTITIES.  43 

In  Algebra,  besides  these,  we  consider  another  set  of  symbols, 
—  1,-2,  —  3,  —  4,  etc.,  and  intermediate  fractions. 

The  relation  between  positive  and  negative  quantities  is  ex- 
hibited to  the  eye  in  the  following  diagram,  where  the  distance 
from  the  zero  point  A  to  any  point  in  the  indefinite  line  BC  is 
considered  positive  or  negative  according  as  that  point  is  on  the 
right  or  on  the  left  of  A : 

Negative.  PoBitive. 


I        I        I        I        I        I        I        I        I        I        I        I       I        I        i       I        I        I       I 
-9-8-7-6-5-4-3-2-1      012     3456789 

109.  In  the  preceding  chapter  we  have  given  rules  for  the 
Addition,  Subtraction,  Multiplication,  and  Division  of  algebraic 
expressions.  Those  rules  were  based  on  arithmetical  notions,  and 
were  shown  to  be  true  so  long  as  the  expressions  represented  pos- 
itive quantities.  Thus,  when  we  introduced  such  an  expression 
as  a  —  h,  we  supposed  a  and  h  to  be  positive  quantities,  and  a  to 
be  greater  than  h.  But  as  we  wish  hereafter  to  include  negative 
quantities  among  the  subjects  of  our  reasoning,  it  becomes  neces- 
sary to  recur  to  the  consideration  of  these  primary  operations. 
Now  it  is  found  convenient  to  have  the  laws  of  the  fundamental 
operations  the  same  whether  the  symbols  denote  positive  or  nega- 
tive quantities,  and  we  may  secure  this  convenience  by  suitable 
definitions. 

110.  The  Absolute  Value  of  a  quantity  is  the  number 
represented  by  that  quantity  taken  independently  of  the  sign 
which  precedes  it.  Two  quantities  are  equal  when  they  have  the 
same  absolute  value  and  are  preceded  by  like  signs.  Two  quan- 
tities may  have  the  same  absolute  value  and  be  unequal.  Thus, 
+  7  and  —  "^  have  the  same  absolute  value,  but  they  are  not 
equal.  Sucn  quantities  as  +  7  and  —  7  are  sometimes  said  to 
be  numerically  equal. 

111.  In  Arithmetic  the  object  of  addition  is  to  find  a  number 
which  shall  contain  as  many  units  as  all  the  given  numbers  taken 
together.  This  notion  is  not  applicable  to  negative  quantities ; 
that  is,  we  have  as  yet  no  meaning  for  the  phrase  "  add  —  3  to 
+  5,"  or  "  add  —  3  to  —  5."    We  shall  therefore  give  a  meaning 


44  POSITIVE     AND     NEGATIVE     QUANTITIES.  ' 

to  the  word  add  in  such  cases,  and  the  meaning  we  propose  is 
determined  by  the  following 

RULES. 

I.  To  add  two  qiiantities  tuith  like  signs,  add  their  absolute 
values,  and  prefix  the  common  sign  to  the  sum. 

n.  To  add  tivo  quantities  with  unlike  signs,  subtract  the  less 
absolute  value  from  the  greater,  and  prefix  to  the  remainder  the 
sign  of  that  quantity  which  has  the  greater  absolute  value. 

Thus,  the  sum  of  3  and  5  is  8 ;  the  sum  of  —  3  and  —  5  is 

—  8 ;  the  sum  of  —  3  and  5  is  2 ;  and  the  sum  of  3  and  —  5 

is  —2. 

112.  That  the  rules  of  the  preceding  Article  are  not  alto- 
gether arbitrary  will  appear  from  the  following  illustrations: 

1.  Suppose  a  man  starts  from  A  in  the  line  BC  (108),  and 
travels  first  3  miles  toward  the  right,  and  then  5  miles  further  in 
the  same  direction ;  his  final  distance  from  A  will  be  8  miles  in 
the  positive  direction.  This  may  be  considered  as  an  intei-preta- 
tion  of  the  8  obtained  by  adding  3  to  5. 

2.  Suppose  a  man  starts  from  A  and  travels  first  3  miles 
toward  the  left,  and  then  5  miles  further  in  the  same  direction ; 
his  final  distance  from  A  will  be  8  miles  in  the  negative  direction. 
This  may  be  considered  as  an  interpretation  of  the  —  8  obtained 
by  adding  —  3  to  —  5. 

3.  Suppose  a  man  starts  from  A  and  travels  first  3  miles 
toward  the  left,  and  then  turns  and  travels  5  miles  toward  the 
right ;  his  final  distance  from  A  will  be  2  miles  in  the  positive 
direction.  This  may  be  considered  as  an  interpretation  of  the  2 
obtained  by  adding  —  3  to  5. 

4.  Suppose  a  man  starts  from  A  and  travels  first  3  miles 
toward  the  right,  and  then  turns  and  travels  5  miles  toward  the 
left ;  his  final  distance  from  A  will  be  2  miles  in  the  negative 
direction.    This  may  be  considered  as  an  interpretation  of  the 

—  2  obtained  by  adding  3  to  —  5. 


POSITIVE    AND    NEGATIVE    QUANTITIES.  45 

113.  In  Algebra,  addition  does  not  necessarily  imply  aug- 
mentation in  an  arithmetical  sense ;  nevertheless,  the  word  sum 
is  used  to  denote  the  result.  Sometimes,  when  there  might  be 
an  uncertainty  on  the  point,  the  phrase  algebraic  sum  is  used  to 
distinguish  such  a  result  from  the  arithmetical  sum  which  would 
be  obtained  by  the  addition  of  the  absolute  values  of  the  terms 
considered. 

114.  In  arithmetical  subtraction  we  have  to  take  one  num- 
ber, which  is  called  the  subtrahend^  from  another,  which  is  called 
the  minuend,  and  the  result  is  called  the  remainder.  The  re- 
mainder, then,  may  be  defined  as  that  number  which  must  be 
added  to  the  subtrahend  to  produce  the  minuend,  and  the  object 
of  subtraction  is  to  find  this  remainder. 

We  shall  use  the  same  definition  in  algebraic  subtraction; 
that  is,  we  say  that  in  subtraction,  we  have  to  find  the  quantity 
which  must  be  added  to  the  subtrahend  to  produce  the  minuend. 

B,  ULE. 

Change  the  sign  of  every  term  in  the  subtrahend,  and  add  the 
result  to  the  minuend  ;  the  sum  thus  obtained  will  be  the  remain- 
der required, 

115.  By  the  rule  of  Art.  114,  the  following  results  are 
obtained : 

1.  Subtracting  3  from  8,  w^e  obtain  5. 

2.  Subtracting  8  from  3,  we  obtain  —  5. 

3.  Subtracting  —  3  from  —  8,  we  obtain  —  5. 

4.  Subtracting  —  3  from  8,  we  obtain  11. 

5.  Subtracting  8  from  —  3,  we  obtain  —  11. 

Let  us  now  recur  to  the  diagram  (108)  and  see  how  these 
results  are  to  be  interpreted. 

1.  Starting  from  the  subtrahend  3,  we  must  move  a  distance 
of  5  toward  the  right — that  is,  in  the  positive  direction — in  order 
to  reach  the  minuend  8 ;  hence,  the  remainder  is  5. 

2.  Starting  from  the  subtrahend  8,  we  must  move  a  distance 
of  5  toward  the  left — that  is,  in  the  negative  direction — in  ordei 
to  reach  the  minuend  3 ;  hence,  the  remainder  is  —  5. 


46  POSITIVE    AND    NEGATIVE    QUANTITIES 

3.  Starting  from  the  subtrahend  ~  3,  we  must  move  a  dis- 
tance of  5  towai'd  the  lefty  in  order  to  reach  the  minuend  —  8 ; 
hence,  the  remainder  is  —  5. 

4.  Starting  from  the  subtrahend  —  3,  we  must  move  a  dis- 
tance of  11  toward  the  right,  in  order  to  reach  the  minuend  8; 
hence,  the  remainder  is  11. 

5.  Starting  from  the  subtrahend  8,  we  must  move  a  distance 
of  11  toward  the  lefty  in  order  to  reach  the  minuend  —  3 ;  hence, 
the  remainder  is  —  11. 

116.  In  the  multiphcation  of  one  monomial  by  another  there 
are  four  cases  to  be  considered. 

1st.  When  the  multiphcand  and  multipher  are  positive. 

2d.  When  the  multiphcand  is  negative  and  the  multiplier 
positive. 

3d.  When  the  multiplicand  is  positive  and  the  multiplier  neg- 
ative. 

4th.  When  the  multiphcand  and  multiplier  are  negative. 

It  was  shown  in  Art.  70  that 

{a  —  h)(c^d)  =  ac  —  ad  —  hc  +  hd     .     .     .     (1) 

Kow,  although  the  result  was  obtained  on  the  supposition 
that  ayh  and  cy  d,  it  will  be  convenient  to  assume  that  (1)  is 
true  for  all  values  of  the  letters.  In  this  way  uniformity  of  re- 
sults will  be  secured. 

Suppose  J  =  0,  and  J  =  0;  then  (1)  becomes 

(a  —  0)(c~0)  =  flc  —  flxO  —  Oxc  +  OxO; 
that  is,  a  X  c  =  ac. 

Suppose  a  =  0,  and  ^  =  0 ;  then  (1)  becomes 
(^^h){^c)  =  -hc. 

Suppose  J  —  0,  and  c  =  0 ;  then  (1)  becomes 

a{—d)  =  —ad. 
Suppose  a  =  0,  and  c  =  0 ;  then  (1)  becomes 

{-h){-d)=hd. 

Hence,  to  multiply  one  monomial  by  another,  we  have  the 
following 


POSITIVE    AKD    NEGATIVE    QUANTITIES.  47 


MULE. 

Multiply  without  considering  the  signs,  and  prefix  -f  or  — 
to  the  2>roducty  according  as  the  two  monomials  have  like  signs  or 
unlike  signs. 

117.  In  division  we  have  the  product  of  two  factors,  and  one 
of  them  given  to  find  the  other.  Therefore,  since  the  product  of 
the  divisor  and  quotient  must  he  equal  to  the  dividend,  we  have 
for  the  sign  of  the  quotient  the  following 

MULE, 

When  the  dividend  and  divisor  have  like  signs,  the  quotient 
must  have  the  sign  +  ;  tuhen  the  dividend  and  divisor  have  un- 
like signs,  the  quotient  must  have  the  sign  — . 

118.  The  words  greater  and  less  are  often  used  in  Algebra  in 
an  extended  sense.  We  consider  a  greater  than  b,  or  b  less  than 
a,  when  a  —  J  is  a  positive  quantity.  This  is  consistent  with 
ordinary  language  when  a  and  b  are  positive  numbers,  and  it  is 
found  convenient  to  extend  the  meaning  of  the  words  greater  and 
less,  so  that  we  may  still  consider  a  greater  than  b,  when  a  or  b 
is  negative,  or  when  both  are  negative.  Thus,  in  algebraic  lan- 
guage, 1  is  greater  than  —  2,  and  —  2  is  greater  than  —  3 ;  for 
1  -I  (—  2)  =  +  3,  and  _  2  —  (-  3)  =  +  1  (114). 

In  this  extended  or  algebraic  setise  a  negative  quantity  may  be 
said  to  be  less  than  zero.  Thus,  —  2  is  algebraically  less  than 
zero;  forO  — (— 2)=  +  2. 

119.  That  a  negative  quantity  is  not  less  than  zero  in  the 
arithmetical  sense  may  be  shown  thus : 

It  is  evident  that  ^  =  ^  (H'^)-  Now,  if  —  1  is  less 
than  zero,  much  more  will  it  be  less  than  -|-  1 ;  that  is,  the  nu- 
merator of  the  fraction  ^  will  be  greater  than  its  denominator; 

hence,  the  numerator  of  the  fraction  ~  will  be  greater  than  its 

denominator ;  therefore,  —  1  is  less  than  +  1  and  greater  than 
-f  1,  which  is  absurd. 


CHAPTEE    lY. 
GREATEST  COMMON  DIVISOR  AND  LEAST  COMMON  MULTIPLE. 


GREATEST    COMMON    DIVISOR. 

130.   A  Common  Divisor  or  Common  Measure 

of  two  or  more  quantities  is  any  quantity  that  will  exactly  divide 
them.  Thus,  a,  J,  and  ab  are  common  divisors  of  ab^  and  abx. 
Any  factor  common  to  two  or  more  quantities  is  a  common  divi* 
sor  of  them. 

121.  Commensurable  Quantities  are  those  which 
have  a  common  divisor.     Thus,  ah^  and  abx  are  commensurable. 

122.  Incom^mensurable  Quantities  are  those  which 
have  no  common  divisor.  Thus,  ab^  and  cdx  are  incommen- 
sunible. 

123.  The   Greatest   Commmi  Divisor  of  two  or 

more  quantities  is  that  common  di\isor  of  them  which  contains 
the  greatest  number  of  prime  factors.  Thus,  Qdh^  is  the  greatest 
common  divisor  of  Via^bx^  and  l%a^cxz. 

For  brevity,  Ave  shall  sometimes  use  G.  C.  D.  for  the  phrase 
greatest  common  divisbVi 

124.  To  find  the  G-.  C.  D.  of  two  or  more  quantities. 

Since  every  factor  of  a  quantity  is  a  divisor  of  that  quantity, 
it  follows  that  all  the  factors  common  to  t^vo  or  more  quantities 
are  all  the  common  divisors  of  those  quantities.  Again,  since  the 
product  of  any  number  of  factors  of  a  quantity  is  a  divisor  of  that 
quantity,  it  follows  that  the  product  of  all  the  factors  common  to 
two  or  more  quantities  is  a  common  divisor  of  those  quantities. 


GREATEST    COMMOlf    DIVISOR.  49 

Moreover,  this  product  is  the  greatest  common  divisor,  for  it  con- 
tains all  the  factors  common  to  the  given  quantities ;  therefore, 
if  another  factor  were  introduced  into  this  product,  the  result 
would  not  divide  at  least  one  of  the  given  quantities. 

Hence,  when  the  given  quantities  can  he  resolved  into  prime 
factors  by  methods  already  explained,  the  G.  C.  D.  may  be  found 
by  the  following 

RULE. 

Resolve  each  of  the  given  quantities  into  its  prime  factors, 
then  the  product  of  all  the  prime  factors  which  are  common  to 
those  quantities  will  ie  the  0.  C.  D.  required. 

EXAMPLES. 

1.  What  is  the  G.  0.  D.  of  ^%x,  Qalh?,  and  IWl^c^da^'i 

Aa%x  =  2  •  2  •  aal)x, 
Qab^a:^=  3  *  2  *  abixxx, 
and  lOaWc^dx^  =  2  •  baahhhccccdxx. 

The  common  factors  are  2,  a,  h.  and  x ;  hence,  2dbx  is  the 
G.  C.  D.  required.  That  ^ahx  is  the  greatest  C.  D.  is  evident; 
for  if  an  additional  factor,  as  a,  be  introduced,  the  product  2a^x 
will  not  be  a  divisor  of  all  the  given  quantities. 

2.  What  is  the  G.  0.  D.  of  ^am^  +  ^bm^  and  Zan  +  Un  ? 

4flm2  4-  ^hm^  =  2  •  2  •  mm  {a  -f  5), 
and  ^an  -f  ^hn  =  3  *  7^  (a  +  &). 

The  only  factor  common  to  both  the  given  quantities  is 
a  -\-  b;  hence,  it  is  the  G.  0.  D.  required. 

Remakk. — When  there  is  only  one  common  divisor,  as  in  the  preceding 
example,  it  would  seem  to  be  improper  to  speak  of  it  as  the  greatest  G.  D. 
Nevertheless,  since  the  common  divisor,  in  such  cases,  is  found  by  the  gen- 
eral rule,  we  shall,  for  the  sake  of  uniformity,  call  it  the  G.  C.  D. 

3.  What  is  the  G.  C.  B.  of  a^  —  y^  ond  a?  —  y^? 

a?-y^={x-y)  (x^  -\-xy  +  y^), 
and  x^  —  y^  =  (x  —  y){x  +  y); 

hence,  x  —  y  ia  the  G.  0.  D. 
4 


50  GREATEST    COMMON    DIVISOE. 

4.  What  is  the  G.  C.  D.  of  aa^ -{-  3ax  +  2a  and  ax^  —  ax  —  2a? 
aa^  +  3ax~{-2a  =  a{x  +  l){x+  2), 
and  ax^  —  ax  —  2a  =  a(x  -\-  1)  (a;  —  2) ; 

hence  a(x  +  1)  is  the  G.  C.  D. 

6.  What  is  the  G.  C.  D.  of  a.-2  —  7a;  +  12  and  a«  —  Sa-  +  15  ? 

Ans.  x  —  3. 

6.  What  is  the  G.  C.  D.  of  a**  — a;—  12  and  apu^—lax  +  12a? 

Ans.  X  —  4. 

7.  What  is  the  G.  C.  D.  of  2a:3  ^  g^.  _  3  and  2a:3  +  2a;  —  24? 

Ans,  2  (a;  +  4). 

8.  What  is  the  G.  CD.  of  2ar»4-  4a;y  +  2y^  and  Zax^-\-^axy-\- 
3a^  ?  Ans,  (x  +  y)  (x  -\-  y). 

125.  It  is  sometimes  rery  difficult,  if  not  impossible,  to  re- 
solve the  given  quantities  into  their  prime  factors  by  inspection. 
We  shall  therefore  proceed  to  demonstrate  the  following  rule, 
which  is  more  general  in  its  application : 

RULE    FOR   FINDING    THE    G.  C.   D.  OF    TWO    ALGEBRAIC 
EXPRESSIONS. 

I.  Let  A  aiid  B  denote  the  two  expressmis  j  let  them  he  ar- 
ranged according  to  the  desceiiding  powers  of  some  common  letter, 
and  suppose  the  exponent  of  the  highest  power  of  that  letter  in  A 
not  less  than  the  exponent  of  the  highest  power  of  the  same  letter 
in  B. 

n.  Divide  A  hy  B  ;  then  make  the  remainder  a  divisor  and  B 
the  dividend.  Again,  make  the  new  remainder  a  divisor  and  the 
preceding  divisor  the  dividend.  Proceed  in  this  way  until  there 
is  no  remainder  ;  then  the  last  divisor  is  the  G.  C.  D,  required. 

The  demonstration  of  the  preceding  rule  depends  upon  the 
following  Lemmas: 

Lem.  I. — If  P  is  a  divisor  of  A,  then  it  will  be  a  divisor  of 
mK.  For,  since  P  is  a  divisor  of  A,  we  may  suppose  A  =  aP; 
then  mA=i  ma?  (42,  4) ;  but  P  is  a  divisor  of  maV ;  therefore, 
since  wA  =  maV,  P  is  a  divisor  of  mA. 


AiB 

pB  p 

B 

0 

^0 

^ 

c 

D 

r\) 

r 

GREATEST    COMMON    DIVISOR.  M 

Lem.  II. — If  P  is  a  divisor  of  A  and  B,  then  it  will  be  a 
divisor  of  mK  ±  nB.  For,  since  P  is  a  divisor  of  A  and  B,  we 
may  suppose  A=«P,  and  B=Z'P;  then  mK±nB={ma±nb)^; 
hence  P  is  a  divisor  of  mA.  ±_  wB. 

Let  A  and  B  denote  the  two  expressions  whose  G.  C.  D.  is  to 
be  found;  let  them  be  aiTanged  according  to  the  descending 
powers  of  some  common  letter,  and  suppose  the 
exponent  of  the  highest  power  of  that  letter  in 
A  not  less  than  the  exponent  of  the  highest 
power  of  the  same  letter  in  B.  Divide  A  by  B ; 
let  p  denote  the  quotient,  and  0  the  remainder. 
Divide  B  by  0 ;  let  5'  denote  the  quotient,  and 
D  the  remainder.  Divide  C  by  D ;  let  r  denote 
the  quotient,  and  suppose  there  is  no  remainder. 

Now,  since  the  dividend  is  equal  to  the  product  of  the  divisor 
and  quotient,  increased  by  the  remainder,  we  have  the  three  fol- 
lowing equations : 

A=i?B  +  C    .    .    .     (1), 

B  =  ^C  +  D    .    .    .     (2), 

C  =  rD    .    .    .     (3). 

We  shall  first  show  that  D  is  «  common  divisor  of  A  and  B. 
D  is  a  divisor  of  C,  since  C  =  rD ;  hence  (Lem.  I),  D  is  a  divi- 
sor of  q(jy  and,  therefore  (Lem.  II),  it  is  a  divisor  of  qO  -\-D\ 
that  is,  D  is  a  divisor  of  B.  Again,  since  D  is  a  divisor  of  B  and 
C,  it  is  a  divisor  of  joB  +  C ;  that  is,  D  is  a  divisor  of  A.  Hence 
D  is  a  divisor  of  A  and  B. 

We  have  thus  shown  that  D  is  a  conmion  divisor  of  A  and  B ; 
we  shall  next  show  that  it  is  their  greatest  common  divisor. 

Equations  (1)  and  (2)  may  be  written  as  follows: 

A-pB  =  Q    .    .    .     (4), 

B-^C  =  D    .    .    .     (5)     (42,3). 

Now,  every  common  divisor  of  A  and  B  is  a  divisor  of  A~joB, 
that  is,  C  (Lem.  II) ;  hence  every  common  divisor  of  A  and  B 
is  a  common  divisor  of  B  and  C.    Similarly,  every  common  divi- 


52  GREATEST    COMMOJS^    DIVISOR. 

sor  of  B  and  C  is  a  common  divisor  of  C  and  D.  "We  have  thus 
shown  that  D  is  a  common  divisor  of  A  and  B,  and  that  every 
common  divisor  of  A  and  B  is  a  divisor  of  D.  But  no  expression 
of  a  higher  degree  than  D  is  a  divisor  of  D.  Therefore,  D  is  the 
G.  C.  D.  required. 

Cor.  1. — Every  common  divisor  of  A  and  B  is  a  divisor  of 
their  G.  C.  D. ;  and  every  divisor  of  their  G.  C.  D.  is  a  common 
divisor  of  A  and  B. 

Cor.  2. — Suppose  we  have  to  find  the  G.  C.  D.  of  A  and  B ; 
and  at  any  stage  of  the  process  suppose  we  have  the  expressions 
K  and  R,  one  of  which  is  to  be  a  dividend  and  the  other  a  divi- 
sor. Let  R  =  ?wS,  where  m  has  no  factor  which  K  has ;  then  m 
may  be  rejected ;  that  is,  instead  of  continuing  the  process  with 
K  and  R,  we  may  continue  it  with  K  and  S.  For,  by  what  has 
been  already  shown,  we  know  that  A  and  B  have  the  same  com- 
mon divisors  as  K  and  R  have.  Now,  any  common  divisor  of  K 
and  S  is  a  common  divisor  of  K  and  R.  Therefore,  any  common 
divisor  of  K  and  S  is  a  common  divisor  of  A  and  B.  Again,  any 
common  divisor  of  K  and  R  is  a  common  divisor  of  K  and  wzS, 
for  ??iS  =  R  But  m  has  no  factor  which  K  has.  Therefore,  any 
common  divisor  of  K  and  R  is  a  common  divisor  of  K  and  S. 
Hence,  A  and  B  have  the  same  common  divisors  as  K  and  S 
have. 

Cor.  3. — A  factor  of  a  certain  kind  may  be  introduced  at  any 
stage  of  the  process. 

Suppose  we  have  to  find  the  G.  C.  D.  of  A  and  B ;  and  at  any 
stage  of  the  process  suppose  we  have  the  expressions  K  and  R, 
one  of  which  is  to  be  a  dividend  and  the  other  a  divisor.  Let 
L  =  wK,  where  n  has  no  factor  which  R  has ;  then  n  may  be 
introduced ;  that  is,  instead  of  continuing  the  process  with  K  and 
R,  we  may  continue  it  with  L  and  R.  For  A  and  B  have  the 
same  common  divisors  as  K  and  R  have ;  and  any  common  divi- 
sor of  K  and  R  is  a  common  di\nsor  of  L  and  R.  Therefore,  any 
common  divisor  of  A  and  B  is  a  common  divisor  of  L  and  R. 
Again,  any  common  divisor  of  L  and  R  is  a  common  divisor  of 
wK  and  R    But  n  has  no  factor  that  R  has.     Therefore,  any 


GREATEST    COMMOiT    DIVISOR. 


63 


common  divisor  of  L  and  E  is  a  common  divisor  of  K  and  E. 
Hence  A  and  B  have  the  same  common  divisors  as  L  and  E 
have. 


Jii  USTJIJLTIONS. 


1.  Find  the  G.  C.  D.  of  x^—6x-^S  and  4:X^—21x'^-\-16x+20. 
The  operation  may  be  arranged  thus : 


4a^  —  21x^-\-15x-\-20 

a;2  _  6a;  4-  8 

4:^3  —  24x2+32^ 

4a; +  3 

3a;2—  17a;  -f  20 
Sx^-  18a;  +  24 

a;8  _    6a;  +  8 
a;8—    4a; 

a;-4 
a;-2 

—  2a; +  8 

—  2a;  +  8 

Hence   a;  —  4  is  the  G.  0.  D.  required. 

2.  Find  the  G.  C.  D.  of  a;2  +  5a;  +  4  and  a^  -{.  4:X^  ^  6x  +  2. 

a;2  +  5a;  +  4 


a^-\-  4a;2  +  5a;  +  2 
x^-i-  5x^  -{-  Ax 


a;-l 


a;2+    a; +  2 
x^  —  5a;  —  4 


a;8  +  5a;  +  4 
a^^-l-  a; 


6x  -\-  6 


4a;  +  4 
4a;  +  4 


a;       4 
6"^  6 


This  example  introduces  a  new  point  for  consideration.    The 

last  divisor  here  is    Gx  -{-  6;    this,  according  to  the  rule,  must  be 

the  G.  0.  D.  required.     When  a;^  +  5a;  +  4  is  divided  by  6a;  +  6, 

X       4 
the  quotient  is  ^  +  - .    If  the  other  given  expression  be  divided 

by  6x  +  6,  the  quotient  will  be  -^  +  «  +  q  • 

O  /i  O 

It  may  at  first  appear  that  6a;  +  6  cannot  be  a  divisor  of  the 
two  given  expressions,  since  the  quotients  contain  fractions.  But 
we  observe  that  in  these  quotients  the  letter  x  does  not  appear  in 


54  GREATEST    COMMON    DIVISOR. 

X      4 
the  denominator  of  any  fraction.    Such  expressions  as  -  +  -  and 

x^       X       \ 

—  -f  5  +  ^  are  said  to  be  entire  with  reference  to  x, 

Xi  At  O 

When  we  say  that  6a;  -|-  6  is  the  G.  C.  D.  of  the  two  given 
expressions,  we  mean  that  no  common  divisor  can  be  found  which 
contains  a  higher  power  of  x  than  6a:  +  6.  Other  common  divi- 
sors may  be  found  which  differ  from  this  so  far  as  respects  numer- 
ical coefficients  only.  Thus,  3a;  +  3  and  2a;  +  2  are  common 
di\nsors.  Again,  a;  +  1  is  also  a  common  divisor,  and  the  corres- 
ponding quotients  are  a;  4-  4  and  a:^  -f-  3a;  +  2.  We  may  then 
conveniently  take  a;  -f  1  as  the  G.  C.  D.,  since  the  quotients  do 
iiot  contain  fractional  coefficients. 

We  may  avoid  fractional  coefficients  by  proceeding  as  in  the 
following  example : 

3.  Find  the  G.  C.  D.  of  Zx^  —  V^t?  +  15a;  +  8  and  a*  — 
2a:*  —  6a;3  ^  4^  ^  13a;  ^  6. 


3a«  — 10a;s+  15a:  +  8 
32«  —  6a;*  —  18a;8  +  12a;2^  39a;  +18 


a;«— 2a:*— 6a*+4a;»+ 13a;+ 6 


G^4-    8a:3_i2a4j_24a;— 10 

Before  proceeding  to  the  next  division,  we  may  reject  the 
factor  2  from  every  term  of  the  new  divisor  (125,  Cor.  2),  and 
multiply  every  term  of  the  new  dividend  by  3  (135,  Cor.  3).  We 
then  continue  the  operation  thus : 

3a:5—  6.x-*— 18a;3  +  12ay^  +  39a;  +  18[3a;^  +  4a;^— 6a;2— 12a:— 5 
3a:5_^   4y4_  6a;3_i2a:2—  5a:  \x  ' 

-^lOa;*— 12a:3^24a;24.44a;^18 

Rejecting  the  factor  2  from  every  term  of  the  last  remainder, 
and  multiplying  the  result  by  3,  we  have  the  expression, 
—  15a:*  —  18a;3  _^  36^^  ^  ee^:  +  27. 
We  then  continue  the  operation  thus : 

—  15a;*  —  18a:3  ^  36^:2  4.  gga:  +  27  I  3a:*  +  4a:^  —  6a:^  —  12a:  —  5 

—  15a:*  —  20y3  +  30a;g  +  60a;  +  25  |  —5 

2a;3^    6ar5+    6a;+    2 


GREATEST    COMMOIT    DIVISOR.  65 

We  now  reject  the  factor  2  from  every  term  of  this  remainder, 
and  continue  the  operation  thus : 

—  03^  —  Idx^  —  Ibx  —  5 

—  5a^  —  15a:=2  —  15a:  —  5 

Hence   o:^  -\- djfi  +  3x  +  1  is  the  G.  C.  D.  required. 

126.  Suggestions.— Suppose  the  given  expressions  A  and 
B  to  contain  a  common  Victor  F,  which  is  obvious  on  inspec- 
tion. Let  A  =  aF,  and  B  =  bF.  Then  F  will  be  a  factor  of 
the  G.  C.  D.  (120).  We  may  then  find  the  G.  C.  D.  of  a  and  b, 
and  multiply  it  by  F ;  the  product  will  be  the  G.  0.  D.  of  A 
and  B. 

In  like  manner,  if  at  any  stage  of  the  operation  we  perceive 
that  a  certain  factor  is  common  to  the  dividend  and  divisor,  we 
may  omit  it  and  continue  the  operation  with  the  remaining  fac- 
tors. The  factor  omitted  must  then  be  multiplied  by  the  last 
divisor  obtained  by  continuing  the  operation  ;  the  product  will  be 
the  G.  C.  D.  required. 

127.  To  find  the  G.  C.  D.  of  three  Algebraic  Ex- 
pressions, A,  B,  and  C. 

Find  the  G.  C.  D.  of  two  of  them,  as  A  and  B.  Let  D  denote 
this  G.  C.  D.;  then  the  G.  0.  D.  of  C  and  D  will  be  the  G.  C.  D. 
of  A,  B,  and  0.  For  every  common  divisor  of  0  and  D  is  a  com- 
mon divisor  of  A,  B,  and  C  (125,  Cor.  1).  Again,  every  com- 
mon divisor  of  A,  B,  and  0  is  a  common  divisor  of  C  and  D. 
Hence  the  G.  C.  D.  of  C  and  D  is  the  G.  C.  D.  of  A,  B,  and  C. 


128.  Find  the  G.  C.  D.  of 

1.  a^  —  Sx-j-2  and  x^  —  x  —  2.  Ans.  x  —  2. 

2.  a:3+3a^i+4a;4-12  and  y^  +  4.x^  +  4.x +  Z.  Ans.  a;  +  3. 

3.  :i^j^y?j^x—^  and  a?+37?-^6x-\-3.  Ans.  x^  f  2a;  +  3. 


56  LEAST     COMMON     MULTIPLE. 

4.  a^  H-  1  and  a^  +  fux^  +  ma;  +  1.  Ans.  x  +  1. 

5.  6a:*  —  laa^  —  20ah:  and  3a^-\-  ax-^  4a\    Ans.  3x  +  4«. 

6.  a;5  —  ^  and  gfl  _  y2^  j^ns.  X  —  y. 

7.  ^a?  —  Ua?  +  23a;  —  21  and  Qa^ -\- t?  —  ^^ -^  21. 

^ws.  3ic  —  7. 

8.  a;*  —  3a;8  ^  22^3  +  a;  —  1  and  a^  —  xi^%x  +  2. 

Ans.  X  —  1. 

9.  a4  _  7a^  4.  8a?«  +  28a;  —  48  and  a:8  —  8a:2  ^  19^;  _  14 

Ans.  X  —  2. 

10.  a;*  —  a:®  +  2arJ  4-  a;  +  3  and  a;*  +  2a;8  —  a;  —  2. 

Ans.  a;^  +  a;  4-  1. 

11.  4a;*  +  9a;8  _j.  22?»  —  2a;  —  4  and  3a;8  +  5a?5  —  a;  +  2. 

Ans.  a;  +  2. 

12.  2a;*  —  12a;8  +  19a;8  —  6a;  +  9  and  4a«  —  18a;2  +  19a;  —  3. 

Ans.  a;  —  3. 

13.  6a;*  +  a;*  —  a;  and  4a;8  —  6a;2  —  4a;  +  3.         Ans.  2x  —  1. 

14.  2a;*  +  lla;®  —  ISx^—  99a;  —  45  and  2a;3  _  7a;2_  45^.  _  21. 

A71S.  2xi  -\-7x-^  3. 

15.  a;s  —  9a;2  ^  26a;  —  24,    x^  —  10a^»  +  31a;  —  30,    and     a^  — 
liar'  +  38a;  —  40.  Ans.  x  —  2. 

LEAST    COMMON    MULTIPLE. 

129.  When  one  quantity  is  divisible  by  another,  the  first  is 
called  a  Multiple  of  the  other.  Thus,  6  is  a  multiple  of  2,  and 
a  J  is  a  multiple  of  d. 

130.  A  Common  Multiple  of  two  or  more  quantities 
is  a  quantity  which  is  divisible  by  each  of  them.  Thus,  12  is  a 
common  multiple  of  2  and  3,  and  20a;y  is  a  common  multiple  of 
2a;  and  6y. 

131.  The  Least  Coimnon  Multiple  of  two  or  more 
quantities  is  that  common  multiple  of  them  which  contains  the 
least  number  of  prime  factors.  Thus,  6  is  the  least  common  mul- 
tiple of  2  and  3,  and  10a;y  is  the  least  common  multiple  of  2a; 
and  5y. 


LEAST     COMMON"     MULTIPLE.  57 

For  brevity,  we  shall  sometimes  use  L.  0.  M.  for  the  phrase 
least  common  multiple. 

133.  To  find  the  L.  C.  M.  of  two  or  more  quantities. 

It  is  obvious  that  the  L.  C.  M.  of  two  or  more  quantities  must 
contain  all  the  factors  of  each  of  them,  and  no  otlier  factors. 
Hence,  when  the  given  quantities  can  be  readily  resolved  into  their 
prime  factors,  the  L.  0.  M.  may  be  found  by  the  following 

JRULE. 

I.  Resolve  each  of  the  given  quantities  into  its  prime  factors. 

II.  Multiply  one  of  the  given  quantities  ly  the  product  of  such 
prime  factors  of  the  other  quantities  as  are  not  found  in  it;  the 
result  will  be  the  L.  C.  M.  required. 

Cor. — If  the  given  quantities  are  relatively  prime  (90),  their 
product  is  their  L.  C.  M.  Thus,  the  L.  C.  M.  of  lab  and  Qcd  is 
4:'Zabcd. 

EXAMPJLES. 

1.  Find  the  L.  0.  M.  of  ^x^y  and  12xtj\ 

Wy  =z3'3'x'X'y,    and    1 2xy^  =  3 '2'2' x' y y; 
hence  the  L.  C.  M.  is  9x^y  x2'2'y  =  ddxy. 

2.  Find  the  L.  0.  M.  of  4«2^,  6a%  and  10^2. 

4a2J2  ^  2  •  2a^^  6a^  =  2  •  da%  and  lOa^^  =  2  '  6a^ ; 
hence  the  L.  C.  M.  is  4^252  x  3  x  5ax^  =  QOa^b^x^ 

It  is  not  necessary,  when  the  given  quantities  are  monomials, 
to  actually  separate  the  literal  parts  into  prime  factors,  since  the 
exponent  of  any  letter  shows  how  many  times  it  occurs  as  a 
factor. 

3.  Find  the  L.  0.  M.  of  a^x  —  2ahx  -f  Vhi  and  a^y  —  %. 
ah^—2alx-\-l^x={a—l)(a—l)x,  and  a^y—¥y={a  +  li){a--l)y\ 
hence  the  L.  0.  M.  is  {a^x  —  2ahx  -f  l^x)  {a  -\-  l)y. 


58  LEAST     COMMON     MULTIPLE. 

4.  Find  the  L.  C.  M.  of  2aWcx,  3a^b(^x%  ^acx,  9cVo,  and 
24fl8.  Ans,  naWc^x^^ 

6.  Find  the  L.  C.  M.  of  l^ax,  40^a;,  and  2baWx^. 

Ans.  400a'Z»5^. 

6.  Find  the  L.  C.  M.  of  a:2  __  3^;  4.  2  and  a^  —  1. 

Ans.  a?  —  %3?  —  x  -{■% 

7.  Find  the  L.  C.  M.  of  a^c  +  ^  and  a^  -  1^. 

Ans.  ahc  —  ciRtx  +  ab^  —  hhc. 

8.  Find  the  L.  C.  M.  of  a^  +  2aJ  +  53  and  a^  —  %db  +  V^. 

Ans.  (a2_^)2. 

9.  Find  the  L.  C.  M.  of  a^  +  V^  and  a^  —  1^, 

Ans.  a^  —  J*. 

10.  Find  the  L.  C.  M.  of  o?  —  x  and   x^  —  1. 

A71S.  a?  —  X. 

11.  Find  the  L.  C.  M.  Of  xz  -\-  yz  and  x^y  -\-  xy\ 

Am.  o?yz  +  xyH, 

133.  It  is  sometimes  very  difficult,  if  not  impossible,  to  re- 
solve the  given  quantities  into  their  prime  factors  by  inspection. 
We  shall  therefore  proceed  to  demonstrate  the  following  rule, 
which  is  more  general  in  its  application : 

RULE    FOR   FINDING    THE    L.  C.   M.  OF    TWO    ALGEBRAIC 
EXPRESSIONS. 

Divide  the  'product  of  the  two  expressions  by  their  G.  C.  D. ; 
n  divide  one  of  the  expressions  by  the  G.  C.  D.  and  multiply  the 
quotient  by  the  other  expression. 

Let  A  and  B  denote  the  two  expressions,  and  D  their  G.  C.  D. 
Suppose  A  =  dD,  and  B  =  Z>D.  From  the  nature  of  the  G.  C.  D., 
a  and  b  have  no  common  factor ;  hence  the  L.  C.  M.  of  A  and  B 

is  abJ).    But  dbJ)  —  -^  =:=xB=:^xA. 

Cor. — If  M  be  the  L.  0.  M.  of  A  and  B,  it  is  obvious  that 
every  multiple  of  M  is  a  common  multiple  of  A  and  B. 


LEAST     COMMOIT     MULTIPLE.  6^ 

134.  Every  common  multiple  of  two  algelraic  expressions  is 
a  multiple  of  their  L.  C  M, 

Let  A  and  B  denote  the  two  expressions,  M  their  L.  C.  M. , 
and  let  N  denote  any  other  common  multiple.  Suppose,  if  pos- 
sible, that  when  N  is  divided  by  M,  there  is  a  remainder,  E ;  let 
q  denote  the  quotient.  Then  E  =  K"  —  gM.  Now  A  and  B  are 
common  divisors  of  M  and  N,  and  therefore  they  are  divisors  of 
E  (125,  Lem.  II) ;  that  is,  E  is  a  common  multiple  of  A  and  B. 
But  E  is  of  loiuer  dimensions  than  M ;  hence  there  is  a  common 
multiple  of  A  and  B  of  lower  dimensions  than  their  L.  C.  M. 
This  is  absurd ;  hence,  there  can  be  no  remainder ;  that  is,  N  is  a 
multiple  of  M. 

135.  To  find  the  L.  C.  M.  of  three  Algebraic  expres- 
sions, A,  B,  and  C. 

Find  the  L.  C.  M.  of  two  of  them,  as  A  and  B.  Let  M  denote 
this  L.  C.  M. ;  then  the  L.  C.  M.  of  M  and  C  is  the  required 
L.  C.  M.  of  A,  B,  and  C. 

For  every  common  multiple  of  M  and  0  is  a  common  multiple 
of  A,  B,  and  0  (133,  Cor.).  Again,  every  common  multiple  of 
A,  B,  and  C  is  a  multiple  of  M  and  C  (134).  Therefore,  the 
L.  0.  M.  of  M  and  C  is  the  L.  0.  M.  of  A,  B,  and  C. 

EXAMPLES, 

136.  Find  the  L.  C.  M.  of 

1.  Qx^  —  x  —  1  and  2x^  -f  3a:  —  2. 

Ans.  (2aj2  4.  3a;  —  2)  {Zx  +  1). 

2.  0^  —  1  and  rr2  -f  a;  —  2.  Ans.  {a?  —  V){x  +  2). 

3.  a;3  __  9^  ^  23a;  —  15  and  x^  —  %x  +  7. 

Ans.  (^3  _  9^  _^  2Zx  —  15)  (x  —  7). 

4.  3a;2  __  5a;  4.  2  and  4^  —  4:X^  —  x -\- 1. 

Ans.  (dx  —  2){^x^  —  Axi  —  x-{- 1). 

5.  (x  +  1)  {x^  —  1)  and  a^  —  1.        Ans.  (a;^  _  1)  (a;  +  1)2. 

6.  a^-\-  'Hx^y  —  xy^  —  2f  and  a^  —  2xiy  —  xy^  +  2y^ 

Ans.  (a^i  —  ?/2)  (a;2  _  4^2), 


60 


LEAST     COMMOiq-     MULTIPLE. 


7.  2z  _  1,  4a:2  —  1,  and  Aa^  +  1.  Ans.  16ar*  —  1. 

8.  a^  —  x,  a^  —  1,  and  a;^  +  1.  Ans,  x{7^  —  1). 

9.  a4>  _  4^2^  (a;  ^  2«)3,  and  {x  —  1d)\        Ans.  {x^  —  4«2)3. 

10.  2)3  __  62;2  4-  11a;  —  6,  a;3  —  9a:2  +  26a;  —  24,  and  a;^  _  8a;2+ 
19a;— 12.  ^w«.  (a;  -  1)  (a;  —  2)  (x  _  3)  (a;  —  4). 

11.  a;2  4.  7a;  ^  10,  a;2  _  2a;  —  8,  and  a^^  4-  a;  —  20. 

Ans,  a;3  _|_  33;2  _  isa;  _  40. 

12.  a2  —  3a J  +  252,  a^-ab  —  2^,  and  a2  _  jz. 

^ns.  a3  _  2ff2^>  -  «Z>2  ^  2&8. 

13.  22?*  —  7a;y  +  32/2,  ^^  _  ^^y  ^  ^y^,  and  a;2  —  bxy  +  6i/2. 

^715.  2a;3  —  lla;2y  +  17a;^2  _  5^3^ 


137. 


SYNOPSIS    FOR    REVIEW. 


r  G.  0.  D. 


L.  0.  M. 


Terms  used 


Special  Rule— Dem. 

General  Rule. — Dem. 
Cor.  1,  Ck)R.  a.  Cor.  3. 


Comtnon  Diusor. 
Q.  C.  D. 
Commenaurdble. 
Incommenaurable. 

Lemma  I— Dem. 
Lemma  U—Dem. 
Application  of  Lemmas^ 


{Introduction  of  Factors, 
Rejection  of  Factors. 


Suggestions  .... 

G.  C.  D.  OP  three  Algebraic  expressions. 


{Multiple. 
Common  Multiple. 
L.C.M. 
Special  Rule— Dem. — Cor. 

General  Rule — Dem. 

Theorem— Dem. 

L.  C.  M.  OF  THREE  Algebraic  expressions. 


CHAPTER   Y. 
-^  FEAOTIOI^S. 

DEFINITIONS  AND  FUNDAMENTAL  PRINCIPLES. 

138.  A  Fraction  is  a  quotient  expressed  by  placing  the 
dividend  over  the  divisor,  with  a  line  between  them.     Thus, 

■J  and  T  are  fractions. 
4  0 

139.  The  Numerator  of  a  fraction  is  the  quantity  above 
the  Une,  and  the  Denominator  is  the  quantity  below  the 
line.  The  numerator  and  denominator  of  a  fraction  are  called 
its  Terms, 

140.  If  the  terms  of  a  fraction  are  integers,  we  may  regard 
the  denominator  as  denoting  the  number  of  equal  parts  into 
which  the  unit  (1)  is  divided,  and  the  numerator  as  denoting  how 
many  of  those  parts  are  expressed. 

141.  A  Fractional  Unit  is  one  of  the  equal  parts  into 
which  the  unit  is  divided.    Thus,  in  the  fraction  t>  the  fractional 

unit  is  T. 

0 

142.  An  integer  may  be  considered  as  a  fraction  with  unity 
for  its  denominator.    Thus,  a  =  zr. 

143.  An  Entire  Quantity  is  one  which  does  not  con- 
tain a  fraction.    Thus,  a  +  b  -^  c  is  an  entire  quantity. 

144.  A  Mixed  Quantity  is  one  which  contains  an  en- 


62  PEACTIONS. 


tire  part  and  a  fractional  part.     Thus,   a-\-b  +  -^  is  a  mixed 
quantity. 

145.  A  Simple  Fraction  is  one  whose  terms  are  entire. 

Thus,  :;  is  a  simple  fraction. 

c  -\-  d  ^ 

146.  A  Complex  Fraction  is  one  which  has  a  fraction 
in  one  or  both  of  its  terms.    Thus,  is  a  complex  fraction. 

147.  A  Compound  Fraction  is  the  indicated  product 
of  two  or  more  fractions.  Thus,  t  X  ;i  X  ;7;  is  a  compound  frac- 
tion. 

148.  A  Froper  Fraction  is  one  whose  numerator  is 

less  than  its  denominator.    Thus,  7  is  a  proper  fraction. 

149.  An  Improper  Fraction,  is  one  whose  numerator 

is  equal  to  or  greater  than  its  denominator.    Thus,  -  and 

are  improper  fractions. 

150.  The  J^ecijrrocal  of  a  quantity  is  the  quotient  ob- 
tained by  dividing  unity  by  that  quantity.    Thus,  the  reciprocal 

of  a  is  - . 
a 

151.  To  multiply  a  fraction  by  an  integer. 

Let  T  be  a  fraction,  and  c  an  integer:  then  y  x  c  =  ^. 
0  °  0  0 

For,  in  each  of  the  fractions  ^  and  -7-  the  fractional  unit 

0  0 

(141)  is  t;  hence,  -r-  is  c  times  t  (140). 

Again,  —  x  c  =  y    For,  the  fractional  unit  in  t  is  c  times 
DC  0  0 

the  fractional  unit  in  7-. 

DC 


DEFINITIONS    AND    PRINCIPLES.  63 


RULE. 

Multiply  the  given  numerator  ly  the  given  integer,  and  divide 
the  product  by  the  given  denominator,  or,  divide  the  given  denomi- 
nator hy  the  given  iiiteger,  and  divide  the  given  numerator  hy  the 
quotient. 

153.  To  divide  a  fraction  by  an  integer. 
a  ^     _  a^ 

For,  T  is  c  times  ^  (151) ;  hence  r-  is  -th  of  t. 
0  DC  ^  be       c  b 

.     .  ac  a 

Again,  -^c  =  j^ 

For,  ^  is  c  times  t;  hence   y  is  -th  of  -7-. 
b  b'  b       c  b 


MULE. 

Multiply  the  given  denominator  by  the  given  integer,  and  di- 
vide the  given  numerator  by  the  product,  or,  divide  the  given  nu- 
merator by  the  given  integer,  and  divide  the  quotient  by  the  given 
denominator. 

153.  The  value  of  a  fraction  is  not  changed  by  multiplying 
or  dividing  both  of  its  terms  by  the  same  quantity. 

It  is  evident  that  if  we  multiply  the  fraction  t  hy  c,  and  then 
divide  the  product  hy  c,  the  resulting  fraction  will  he  equal  to  the 
given  fraction.    Now  ^  x  c  =  -r-  (151),  and  "t  "^  ^  =  l^  (153) ; 

,  a      ac        ac      a 

hence  t  =  i->  or  ^  =  t  . 

b      be        be      b 


64  FRACTIONS. 


REDUCTION    OF   FRACTIONS. 

154.  A  fraction  is  in  its  Lowest  Terms  when  its  terms 
have  no  common  factor. 

155.  To  reduce  a  fraction  to  its  lowest  terms. 

RULE. 

Divide  both  terms  of  the  fraction  by  their  G.  C.  D. 
Or,  Resolve  both  terms  of  the  fraction  into  their  prime  factors, 
and  then  cancel  those  factors  ivhich  are  common. 

ILZU8TBATION8, 

lOacx^ 

1.  Reduce  ^,,  ' ,  to  its  lowest  terms. 

The  G.  C.  D.  of  the  terms  of  this  fraction  is  5caf, 
Dividing  both  terms  by  this,  we  "have 

lOaca?^  _  2a 
15bca^  ~  3bx' 

2.  Reduce  r-5 — ^-^  to  its  lowest  terms. 

3a^  —  3ab 

3a^  +  Sab  _  da  {a  +  b)  _  a_+b 
da^  —  'Sab~3a{a  —  t>)  ~~  a—b' 

^   _   ,         6a:2_    7a;_20  ^    .,    ,         ,  , 

3.  Reduce  -r-5 — 7^ ?  to  its  lowest  terms. 

4^  —  27a;  4-    5 

Here  the  G.  C.  D.  of  the  numerator  and  denommator  is 
2x  —  5.    Dividing  both  terms  of  the  fraction  by  this,  we  have 

Qa^—    7a;  — 20_       3a;  +  4 

4a:3_27a;+    5  ~"  2a;3  _^  5^;  _  1 ' 

156.  To  reduce  a  fraction  to  an   entire   or  mixed 
quantity. 

a^-\-ab  ^     a^+b  b  ,     a^—  b  b 

— ■ =  a  -\-  b,    — ■ —  =  a  -f  - ,    and =  a . 

a  'a  a  a  a 


BEDUCTIOl^    OF    FRACTION'S.  65 


^  MULE, 

Divide  the  numerator  hy  the  denominator,  expressing  any  term 
of  the  quotieMdn  a  fractional  form  when  the  divisio7i  cannot  de 
exactly  performed.^ 

157.  To  reduce  an  entire  quantity  to  the  form  of 
a  fraction  having  a  given  denominator. 

Let  it  be  required  to  reduce  x  -\-  y  to  the  form  of  a  fraction 
whose  denominator  shall  he  x  —  y. 

x^y  =  ^  +  ^  =  (^  +  y)  (a^ - ;/)  ^  x^ - y\ 
^  1  X  —  y  X  —  y  ' 

RULE. 

Consider  the  entire  quantity  as  a  fraction  whose  denominator 
is  unity  ;  then  multiply  hotli  terms  of  this  fraction  iy  the  given 
denominator, 

158.  To  reduce  a  fraction  to  an  equivalent  one 
having  a  given  denominator. 

Let  it  be  required  to  reduce  the  fraction  ^  to  an  equiva- 
lent one  having  the  denominator  a^  —  h\ 

Dividing  o^  —  W-  by  a  —  h,  the  quotient  is  a-^-h. 

Multiplying  both  terms  of  the  fraction  7  by  this  quo- 
tient, we  have 

a^-b  _  {a  ■\-b){a-\-  h)  _  {a  +  hf 
a  —  b"         a^  —  l^        -  a^  —  yi' 

RULE. 

Multiply  both  terms  of  the  given  fraction  by  the  quotient  ob- 
tained by  dividing  the  denominator  of  the  required  fraction  by 
the  denominator  of  the  given  fraction, 

159.  To  reduce  a  mixed  quantity  to  the  form  of  a 
fraction. 

Let  it  be  required  to  express  a  +  -  under  a  fractional  form. 


66  FRACTIONS. 

h      ac  -\-h 

a;  +  -  = . 

c  c 

For,  if  we  reduce  the  fraction to  a  mixed  quantity, 

c 

we  obtain  a  -\-  -  (156). 
c 

In  like  manner  we  may  show  that  a = . 

•^  c  c 

RULE. 

Multiply  the  entire  part  hy  the  denominator  of  the  fractional 
part ;  the7i  add  the  numerator  to  the  product,  or  subtract  it  from 
the  product,  according  as  the  fraction  has  the  sign  +,  or  the 
sign  —,  prefixed  ;  the  result  will  he  the  numerator,  and  the  given 
denominator  will  be  the  denominator  of  the  required  fraction. 

160.  To  reduce  fractions  to  equivalent  ones  having 
a  common  denominator. 

a    c  e 

Let  ^,  -j,  and  ^  be  the  proposed  fractions.     If  we  multiply 

both  terms  of  each  of  these  fractions  by  the  product  of  all  the  de- 
nominators except  its  own,  the  values  of  the  fractions  will  not  be 
changed  (153).  Moreover,  the  denominators  of  the  new  fractions 
will  be  equal,  since  each  is  the  product  of  the  denominators  of 
the  given  fractions. 

^,        a      adf    c       bcf         ,    e       bde 


RULE. 

Multiply  both  terms  of  each  of  the  given  fractions  by  the  pro- 
duct of  all  the  denominators  except  its  own, 

161.  To  reduce  fractions  to  equivalent  ones  having 
the  least  common  denominator. 

Let  — ,  — ,  and  —  be  the  proposed  fractions.   The  L.  0.  M. 
nix   my  mz  ^    ^ 

of  the  denominators  is  mxyz,    Now  reduce  each  of  the  given  frao- 


COMBINATIOlfS    OF    FEACTIOKS.  67 

tions  to  an  e^ivalent  one  having  mxyz  for  its  denominator 
(158) ;  tlie  resulting  fractions  are 

ayz  hxz  ,        cxy 

mxyz'      mxyz'  mxyz' 

Now  since  mxyz  is  the  least  quantity  that  can  be  divided  sep- 
arately by  mar,  my,  and  mz,  it  follows  that  the  given  fractions 
have  been  reduced  to  equivalent  ones  having  the  least  common 
denominator. 

Divide  the  L.  C.  M.  of  all  the  denominators  hy  each  denomina- 
tor separately ;  then  multiply  both  terms  of  each  fraction  by  the 
corresponding  quotient. 

ScH. — Before  commencing  the  operation,  each  fraction  must 
be  in  its  lowest  terms. 

COMBINATIONS    OF    FRACTIONS. 

162.  To  find  the  sum  of  given  fractions. 

a   c 
1.  Let  it  be  required  to  find  the  sum  of  the  fractions  t?  t?  and 

^ .    Here  the  given  fractions  have  a  common  denominator.     In 

the  first  fraction  the  fractional  unit  ^  is  taken  a  times ;  in  the 

0 

second  it  is  taken  c  times ;  and  in  the  third,  d  times ;  hence,  in 

the  snm  o 
therefore, 


the  snm  of  these  fractions,  t  must  be  taken  (a  +  c  -t  d)  times ; 


a      £  ^_  a  -\-  c  -^  d 

h^b'^b-         b 

ft  p 
2.  Let  it  be  required  to  find  the  sum  of  the  fractions  ^,  ^,  and 

^.  Here  the  given  fractions  have  unequal  denominators.  Re- 
ducing them  to  equivalent  fractions  having  a  common  denomina- 
tor (160),  we  have 

a      c       e  _adf       M.  jl-^  —  0^^/"+^^/+  ^^^ 
b^d'^f-bdf'^bdf^bdf-  bdf 


68  FBACTIONS. 


n  ULES, 


I.  If  the  given  fractions  have  a  common  denominator,  form  a 
fraction  tvhose  numerator  is  the  sum  of  the  given  numerators, 
and  whose  denominator  is  the  given  common  denominator  j  this 
fraction  luill  he  the  sum  of  the  given  fractions. 

II.  If  the  given  fractions  have  not  a  common  denominator,  re- 
duce them  to  equivalent  ones  having  a  common  denominator  ;  then 
proceed  as  directed  in  L 

163.  To  find  the  diflference  between  two  fractions. 

c  fl 

1.  Let  it  be  required  to  subtract  -r  from  t-  The  fractional 
unit  T  is  taken  a  times  in  the  minuend,  and  c  times  in  the  sub- . 

0 

trahend ;  hence,  it  must  be  taken  {a  —  c)  times  in  the  remain- 
der; therefore, 

a      c  _a  —  c 

h~h~~h~ 

c  a 

2.  Let  it  be  required  to  subtract  -^  from  t. 

Reducing  these  fractions  to  equivalent  ones  having  a  common 
denominator,  we  have 

a      c  _ad      Ic  _ad  —he 
b      li~  hd      hd~      hd 


RULES. 

I.  If  the  given  fractions  have  a  common  denominator,  form  a 
fraction  whose  numerator  is  the  remainder  ohtained  hy  suhtr act- 
ing the  numerator  of  the  suhtr ahend  from  that  of  the  minuend, 
and  luhose  denominator  is  the  given  common  denominator ;  this 
fraction  will  he  the  difference  required. 

IL  If  the  given  fractions  have  not  a  common  denominator,  re- 
duce them  to  equivaleiit  ones  having  a  common  denominator  ;  then 
proceed  as  directed  in  L 


COMBINATION'S    OF    JRACTIOIJS.  69 

164.  To  find  the  product  of  given  fractions. 

CL  C 

Let  it  be  required  to  find  the  product  of  t  and  -^.  The  fol- 
lowing is  usually  given  as  a  solution : 

Put  Y  =  ^>  and  -^  =  w. 

Then  a  =  bm,  and  c  =  dn. 

Hence,  ac  =  bmdn  =zhd  x  mn;    or,  dividing  both  members 

dC 

by  bd  (42,  5),  we  have  j-j  =  mn. 

This  process  is  satisfactory  when  m  and  n  are  really  integers, 
though  under  a  fractional  form,  because  then  the  word  multipli' 
cation  has  its  common  meaning.  It  is  also  satisfactory  when  one 
of  them  is  an  integer,  because  we  can  speak  of  multiplying  a  frac- 
tion by  an  integer,  as  in  Art.  151.  But  when  both  m  and  n  are 
fractions,  we  cannot  speak  of  multiplying  one  of  them  by  the 
other  without  defining  what  we  mean  by  the  term  multiplication  j 
for,  according  to  the  ordinary  meaning  of  this  term,  the  multiplier 
must  be  an  integer. 

The  following  definitions  will  show  more  clearly  the  connec- 
tion between  the  meaning  of  the  word  multiplication  when  ap- 
plied to  integers,  and  its  meaning  when  applied  to  fractions. 
When  we  multiply  one  integer,  a,  by  another,  b,  we  may  describe 
the  operation  thus : 

What  tve  did  with  unity  to  obtain  b,  we  must  now  do  with  a  to 
obtain  b  times  a. 

CL  C 

Now,  let  it  be  required  to  multiply  t  hy  ^.  Adopting  the 
definition  just  given,  we  may  say  that,  what  ive  did  with  unity  to 

obtain  -^,  we  must  now  do  with  ^  to  obtain  the   product   of  ^ 

c  c 

and  -^ .     To  obtain  -^  from  unity,  we  divide  it  into  d  equal  parts, 

and  multiply  one  of  the  parts  by  c ;  therefore,  to  obtain  the  pro- 
duct of  T  and  ^,  we  divide  t  into  d  equal  parts,  and  multiply 


70  FRACTIONS. 

one  of  them  by  c.     Now  ^  -^  J  =  r^  (152),  and  -^-7X^  =  ^-7 
•^  0  Id  ^        '  bd  bd 

(151). 

We  may  therefore  give  the  following  extended  definition : 

Multiplication  is  the  process  of  finding  a  quantity  having  the 
same  relation  to  the  multiplicand  that  the  multiplier  has  to  unity. 

RULE. 

Form  a  fraction  whose  numerator  is  the  product  of  the  given 
numerators,  and  whose  denominator  is  the  product  of  the  given 
denominators  ;  this  fraction  will  he  the  product  required. 

ScH.  1. — This  rule  embraces  all  the  cases  in  which  a  fraction 
is  a  factor.  Thus,  if  it  be  required  to  multiply  a  fraction  by  an 
entire  quantity,  the  latter  may  be  considered  as  a  fraction  whose 
denominator  is  unity  (142). 

ScH.  2. — If  any  factor  is  a  mixed  quantity,  it  is  best  to  reduce 
it  to  the  form  of  a  fi*action  before  commencing  the  operation. 

ScH.  3. — if  the  numerator  and  denominator  of  the  product 
have  any  common  factor,  it  should  be  canceled.     Thus, 
2a2  (a-\.bf  _   2fl2(rt  +  *)2  _  2a^{a  +  b){a-\-b)  _     a  +  b 


ai  —  l^-     4a26         {a'^—l^)^ib      U^b{a  +  b){a-b)      2b(a-b) 
(155). 

165.  To  find  the  qtiotient  of  two  fractions. 

a  c 

Let  it  be  required  to  divide  t  1^7  3  •    Denoting  the  quotient 

by  X,  we  have 

a      c  _ 

V'^'d^'^' 

But  the  product  of  the  divisor  and  quotient  is  equal  to  the 

dividend;  hence, 

c      a 

Multiplying  both  members  of  this  equation  by  -  (42,  4),  we 

c 

have  c      d      a      d 

d      c      0      c 


COMBIITATIOKS    OF    FEACTIOKS.  71 

Canceling  common  factors  (155),  we  have 

a      d 
0       c 

that  is,  the  quotient  is  equal  to  the  product  obtained  by  multiply- 
ing the  dividend  by  the  divisor  inverted. 

nULE. 

Multiply  the  dividend  by  the  divisor  inverted. 

Cob.  1. — The  product  of  a  quantity  and  its  reciprocal  is  unity. 

Thus,  a  X  -  =  1. 
a 

Cob.  2. — To  divide  by  a  quantity  is  the  same  as  to  multiply 
by  its  reciprocal ;  and,  conversely,  to  multiply  by  a  quantity  is 
the  same  as  to  divide  by  its  reciprocal.    Thus, 

« -^  Z*  =  a  X  T  J    and    a  x  J  =  a  ~  t  • 

0  0 

166.  In  the  present  chapter  we  have  thus  far  supposed  each 
letter  to  represent  an  integer ;  but,  by  virtue  of  our  extended 
definitions,  it  may  be  shown  that  all  the  rules  and  formulae  given 
are  true  when  any  letter  represents  a  fraction.    For  example,  let 

it  be  required  to  show  that  t  =  t-  when   a  =  — ,  J  =  — ,  and 
^  b       he  n  q 

r 
c  =  -. 

s 


a 

m 
n 

m 
"~  n 

P 

mq 

~  np* 

ac  = 

_m 

~  n 

r 
X  - 

s 

mr 
-  ns' 

q       s        qs 

,  ac      mr       pr      mr       qs       mrqs      mq 

hence,        t-  = J—^=  —  x  -^^  =  — —  =  — ^• 

oc       ns        qs        ns       pr       nspr       np 


72  FRACTIOKS. 


THE    SIGNS    OF    FRACTIONS. 

167.  Each  sign  in  the  numerator  and  denominator  of  a  frac- 
tion affects  only  the  term  to  which  it  is  prefixed.     Thus,  in  the 

fraction  ^,  the  sign  of  a  is  +,  that  of  J  is  — ,  that  of  c  is 

+,  and  that  of  d  is  — . 

168.  TJie  dividing  line  of  a  fraction  answers  the  purpose  of  a 
vinculum ;  that  is,  it  connects  the  terms  which  the  numerator 
and  denominator  may  each  contain.  Therefore  the  sign  prefixed 
to  the  dividing  fine  affects  the  fraction  as  a  luhole, 

169.  If  the  sign  prefixed  to  the  dividing  line  be  changed, 

the  sign  of  the  fraction  will  be  changed.    Thus,  -r-  =  fl^ ;  but 
ah 

170.  If  the  sign  of  each  term  of  the  numerator  be  changed, 

the  sign  of  the  fraction  will  be  changed.     Thus, ^ —  =  a  —  c ; 

^  ,   —ah  -^-Ic 

but  7 =  —  a  +  c. 


171.  If  the  sign  of  each  term  of  the  denominator  be  changed, 
ah 


the  sign  of  the  fraction  will  be  changed.    Thus,  -7-  =  a ;  but 


172.  We  may  sum  up  the  three  preceding  Articles  thus: 

If  the  sign  prefixed  to  a  fraction,,  or  the  sign  of  each  term  of 
the  numerator,  or  the  sign  of  each  term  of  the  denominator,  he 
changed,  the  sign  of  the  fraction  will  he  changed. 

CoE. — If  any  two  of  these  changes  be  made  at  the  same  time, 
the  sign  of  the  fraction  will  not  be  changed. 

173.  Tlie  Apparent  Sign  of  a  fraction  is  the  sign  pre- 
fixed to  the  dividing  hne  of  that  fraction.     The  Meal  Sign 


EXAMPLES.  73 

of  a  fraction  is  the  sign  of  its  numerical  value.    Thus,  the  ap- 

parent  sign  of  the  fraction ^^—  is  —  ;  but,  if  «  =  3,  5  =  4, 

and  c  =  5,  the  real  sign  is  -f . 

1*74.  EXAMPLES, 

Simplify  the  following  fractions,  from  1  to  12,  inclusive : 
a;2+2a;  — 3  .       a;  +  3 


^'  x^-{-6x  —  1l' 

a;  +  7 

^      X2-3X-4: 

a;  — 5 

a^-6a^ -\.llx  — 6 
a;2  _  3a;  +  2       * 

-47i5.  a;  — 3. 

a^ -{- 3a^  +  Sal?^ -{- b^ 

^ws.  a-{-b. 

a;*  +  10a;3  ^  35^  ^  50^  +  24 
^  +  9a;2  +  26a;  +  24 

Ans,  X  -\-l. 

3a:3_i6^^23a;-6 
•  "iT^  —  Wx^^Yix  —  ^' 

A       3a;— 1 

Ans.  ^ r. 

2a;  —  1      ^ 

6a;3  —  5a;2  4-  4 
^x?  —  x?—x^%' 

.       3x-  +  2 
Ans,  — i—-. 

X+1 

2a;3  4.9^^    ^x  —  3 
•  3a;3^5^2_i5a;^4' 

.       2a; +  3 
^^^•3^-4- 

1  — a;2 

^''''1+x' 

5a^-\-5ax 

Ans.  . 

a  —  x 

a;^  +  2a;2  +  9 
•  ari_4a:3^4^_9- 

"a;2  +  2a;  +  3 

^g    a;2  4-  («  -f  c)  a;  +  ac 
*  a;2  +  (6  +  c)x+  be' 

Ans.^'^l 
X  +  b 

74  FKACTIONS. 

Perform  the  additions  and  subtractions  indicated  in  the  fol- 
lowing examples,  from  13  to  28  inclusive : 

a  +  ^      a— J  a^  —  W 

14.    ^^ r-7  +  -7 jr-  AtlS.    ^. 

2a  —  2^      2^  —-  2a  2 

^„    2  3  2rr  — 3  ^  9 


a;      2a;  — 1       4.^2— 1  *       (4a;2_i)^ 


17. rr  —  -, rrr7..  AuS. 


2a 
Ans.  — . 
n 


18. 


x  —  l       ic-f  2       (a;-f-2)2*  "*  (a;  —  1)  (a;  +  2)2' 

5 1 24 

2(a;+l)       10(.'?;  — 1)       5(2a;  +  3)' 

.  2a:  —  3 

Ans. 


(a;2_l)(2a;  +  3)' 


h—a      a  —  2b^   Sx(a  —  b)  .       ax  —  1^ 

^^'  x-b       x-\-b^    a^-lP  '^^^'  w^r^' 

_^    3  4-  2ar      2  —  3a;      IGa;  —  x^,  1 

20.  -r jr— 1 r —  .     Ans. 


2  —  x         2  +  a;'     a-2  —  4  a;4-2 

oi         3                7           4-20a;  .        ^ 

^^•nr^-rT2^-4^rrT-  ^^^- ^• 

22.4-.  +  ^^--.^,.  Ans.     '^^ 


a  +  b  '   cfi  —  V^      a-  +  V>'  a*  —  b*' 

o„        1  1 1  .       x'  —  ixy  —  y- 

^^-  a:^  _  2,3  +  (.^  +  yf       (X  -  y)2-  ^'**-      (a^  -  y2)» 

24.  i2i±^-?-*-2.  Ans.      "" 


ab(a  —  by      b      a  {a  —  bf 

^K        fl       ,      3a            2«a:  .  4a 

25. \ ; -.  Ans. 


a  —  x       a -{- X      a^  —  7?'  '  a -j- x' 

q^    3a  —  45       2a  —  b  —  c       15a  —  4c      a  —  Ab 


,       Sla  —  ib 
^»*-  -ST— 


EXAMPLES.  75 


27  ^  +  ^ L         ^  +  g  ,  c  +  a 


{h  —  c){c  —  a)       {c  —  a){a^  b)       {a  —b)  (b  —  c)' 

Ans.  0. 
a^  —  be  b^  —  ac  &  —  ab 


(a  +  6)  (a  +  c)       (^>  +  c)  (^  +  a)       (c  +  «)  (c  +  ^')  * 

^4ws.  0. 

29.  Multiply  -5^ r-^-  by  -^ tt-  ^w5.  ^ .-(-. 

^  -^     a  4-  6       ^  x{a  —  b)  {a  +  b)x 

30.  Multiply  ^i^  by  _?!zilL.  ^^^.  -^^^^ . 

o.    T»r  IX-  1    .       XI        3«a;     «2_/p2     Jc  +  ^a;         ,   c  — a; 

31.  Multiply  together  ^,    ^-^,    -^^-^^,  and  --. 

3a; 

^72S.   -7-. 

32.  Prove  that 

33.  Multiply  together  y^y ,    ^^,  and   1  +  j-^. 

Ans.  --. 

X 

34.  Multiply  -^ — —- ^  by     „    \^  — '—' 

^  "    a^  -\-2ax  +  x^     ^    a^—  2ax  +  x^ 

A  c^x 

Ans.  -^ -„. 

a^  —  x^ 

OK    o-      vc  a^  —  b^  a^-b  .       a^  +  b'^ 

35.  Smphfy  ^^^-^^  X  ^^^^.  Ans.  —-. 

36.  Simplify  (^+l-^:zl_  JyL\ ^+1.        Ans.  3. 

^    '^   \x  —  y      X  -{■  y      a^  —  yv    2y 

37.  Simplify  ^-^3  -  ^-^  -  (^^^:py,j. 

«2  _  ^J  _|_  J2 
^''^-    ^2  +  «^,  +  J3 

38.  Multiply  -'--  +  1   by  ^  +  ^  +  1.    ^^z-?.  ^  H- -' +  1. 


76  FRACTIONS. 

39.  Multiply  ir2  — a; +1   by  1  +  -  +  1.        Ans.  x^  +  l-}- ^. 

40.  Simplify  -^ )     ;    ,(   |— t  X  -^ Tn.      Aiis,  )    ^  ,(-,. 

42  Divide  ^(^'^^^)   by  -^^  Ans  l^^^- 

43.  Divide  -^^  by   -f--  ^w^-  zs ^^"a- 

44.  Divide  '-^  +  -^ r by  ^,. 

a;  +  y        a;  —  ^       o?  —  'f     ''    x?  —  y^ 

Ans.      ^ 


45.  Simplify   g  +  ^).(j-J  +  i).  Ans.^-±y 

46.  Simplify  (-^  +  —j)  -  (-^ ^—j).      Ans.  1 

47.  Simplify   (^±^  +  ?)  -  (^-±^ ^-) .         ^^..  1 

^     -^    \x-\-y       yl       \     y  x  +  yj 

48.  Divide   x^ j  by  x  •\-  -.  Ans.  o^  —  x  A „ 

x^      •^  X  X       x^ 

49.  Divide  0?  ^  \  +  2  by  x+^,  Ans.  — - 

x^  -^  X  X 

50.  Divide  x^^l^^   by   ^  -  1+  a;.         ^tjs.  ^^  +  ^  +  ^ 


a;-*      "    X  X 

51.  Divide  fl2_^_^^2Z'c  by  ^-±4-f^. 

Ans.  a^  —  b^  A-  (^  +  ^cic. 

62.  Dmde  — ^- — ^— r — ^^—  by  -oV"  — f— 2- 

«  +  a; 
•  x  —  y 


53.  Diyide  a^  —  h^  —  c^  —  2hc  by 


EXAMPLES.  77 

a  -\-  h  -\-  c 


a  -\-  b  —  c' 

A71S.  a^  _  J2  _|_  ^2  _  2ac. 


54.  Divide  x^  -  Sr-  -  2a^  +  ^^fV  by  Sx  -  Ga -^. 

a;2  +  36?a;  —  2a^ 


Ans, 


X  -\-  6a 


5o.  Divide  zr^  —  4  H — ^   by  ^r .  Ans.  

2a^  x^      -'    2a       X  ax 

a  +  h      a  —  h 

t^n    cs'      ^'c^  ^  +  d      c  —  7l  .       ac  —  M 

56.  Simplify  7 t*  ^ns.  --^-^. 

^    -'   a  +  b      a  —  b  ac  -\- bd 

c  —  d      c  +  d 
a  ■\-  X      a  —  x 

57.  Simplify .  Ans.  —-: . 

^     ''  a  -\-  X      a  —  x  2ax 

a  —  x      a  -{-  X 

a—1       b—1      c—1 

^o    o-      vn  ^abc  a  b  c 

58.  Simplify 


be  +  ca  —  ab  1,1       1 

be  -\-  ac  -\-  ab 


a      b       c 


Ans. 


be  -^  ac  —  ab 

Ko   Q-      r^  (a  +  b      a^-^b^\       (a-b      a^  -  b\ 
59.  Simplify  (^-^  +  -^-^^j  -  (— ^-  -  ^3-^^). 

a^  +  aW  +  ^* 


^^5. 


«^>  (a  -  bf 
60.  Simplify  (^-^^  -  -^,-^3)  ^  (--^  +  -,_-^-^). 


bc(b-cf 
Ans. 


ei.Si.piiiy(J±|-5^:).g^^-^D. 


Ans.        ^ 


78  FRACTIONS. 

62.  Simplify  (i:+*  +  ^)  ^  (^  _  ^) . 

63.  Simplily  — i ^ x  ^ — =.  ^ns.  m, 

n       m 

X  -{-  a      x  —  a 

64  Simplify ■ — .    Ans.  -, 

''x  —  a      X  -\-  a      X  -\-  a      x  —  a  x^  —  a 

x  —  a      X  -{■  a 


65.  Simplify   ^ ^(l  +  — ^^— )• 


a 

b  +  c 

fifi 

bimplify 

1 

X-\- 

1 

^  +  3- 

±_l 

—  X 

07. 

Simplify 

h-¥ 

a 
c 

A 

Ans. 


d(x^l) 


'-J 


CO  —  bv 

68.  Find    the    value    of    ax  +  5?/,    when    x  =  -^^ — -    and 

«r  —  c» 

y= r^.  Ans.  c. 

^       aq  —  bp 

69.  Find  the  value  of ^r-  -\ -rj ,  when  x  = 


x  —  2a      X  —  2b'  a  -\-  y 

Ans.  2. 

70.If|  +  |=l,showthatf-|=J-J. 

a       c^       fv^        (^  n       c 

71.  If  T  +  :7  =  72- ^>  sliow  that  t >  =  1. 

0      a      If^       (P  b      d 


I 


SYNOPSIS    rOR    REVIEW. 


79 


175. 


SYNOPSIS    FOR    REVIEW. 


CHAPTER  V. 
FEAOTIONS. 


f  Denominator.  )  rn,    ^  ,    ,     ,. 

,  ^  ,         >  The  terms  of  a  fraction. 

Numerator.    S 


'  Terms  used  . .  J  ^rrac.  unit. 

Entire  quantity. 
I  Mixed  quantity. 


Simple  Frac. 


Proper. 
Improper. 


To  REDUCE  . . 


Complex  Fractions. 

Compound  Fractions. 

Value  of  fraction  not  changed,  when. 

Fraction  in  its  lowest  terms,  when. 

Fraction  x  Integer.    Bute. 

Fraction  -t-  Integer.    BuZe. 

Fraction  to  mixed  qwintity.    Rule. 

Entire  quantity  to  fraction  having 
given  denominator.    Rule. 

Fraction  to  fraction  having  given  de- 
nominator.   Rule. 

Mixed  quantity  to  form  of  fraction. 
Rule. 

Fractionsto  common  denominator.  Rule. 

Fractions  to  least  com.  denom.    Rule. 

Addition.    Investigation  for  rule.    Rule. 

Subtraction.    Investigation.    Rule. 

Multiplication.    Def.    Investigation.    Rule. 

Division.    Investigation.    Rule.    Cor.  1,  3. 

^gn  prefixed  to  a  term  of  numerator 

or  denominator. 
Sign  prefixed  to  fraction. 
Signs  of      J  Methods  of  changing  sign  of  fraction. 
Fractions.      |  Changes  of  sign  not  affecting  sign  of 
fraction. 
Apparent  sign  of  fraction. 
^  Real  sign  of  fraction. 


CHAPTEE   VI. 

DEFiraiOIfS  AND  GENERAL  PRINCIPLES  RELATING  TO  EQUATIONS. 


176.  An  Equation  consists  of  two  expressions  connected 
by  the  sign  of  equality.     Thus,  x  -\-  a^=m  -\-  n  is  an  eqnarion. 

The  First  Member  of  an  equation  is  the  quantity  on  the  left 
of  the  sign  of  equality,  and  the  Second  Member  is  the  quantity  on 
the  right  of  the  sign.  Thus,  in  the  equation  x  -\-  a  =z  m  -\-  n, 
ic  -f  a  is  the  first  member,  and  m  -\-  n  the  second  member. 

177.  An  Identical  Equation^  or  An  Identity ,  is 

an  equation  whose  members  are  either  identical,  or  may  be 
made  identical  by  performing  the  indicated  operations.  Thus, 
a^  —  x^  ,  a  ax 


ax  -^  b  =:  ax  -^  b, —  =  a  +  x,     and     a      ^ 

^  ^    '      a  —  x  ^    '  1  +  a:       1  +0; 

are  identities. 

178.  It  follows,  from  the  definition,  that  the  members  of  an 
identity  are  equal  for  all  values  that  may  be  assigned  to  each 
letter  which  it  contains. 

Thus  far  the  student  has  been  almost  entirely  occupied  with 
identities.  Thus,  the  equations  given  in  Articles  71,  73,  and 
73  are  identities. 

179.  An  Equation  of  Condition  is  one  whose  mem- 
bers are  equal  only  for  a  limited  number  of  values  of  each  letter 
which  it  contains.  Thus,  x  -\-  1  =z  7  is  an  equation  of  condition, 
because  its  members  are  not  equal  unless  x  =  6. 

An  equation  of  condition  is  called  briefly,  an  equation. 

180.  An  Unknown  Quantity  is  a  letter  to  which  a 
particular  value  or  values  must  be  given  in  order  that  the  mem- 
bers of  an  equation  may  become  identical.    The  equation  is  said 


DEFINITIONS.  81 

to  be  Satisfied  for  such  particular  value  or  yalues.  Thus,  in  the 
equation  x^  —  ^x  =  —  3,  the  unknown  quantity  is  x,  and  when 
a;  =  3  or  1,  the  equation  is  satisfied. 

181.  An  Unknown  Term  of  an  equation  is  a  term  con- 
taining an  unknown  quantity. 

183.  A  Moot  of  an  Equation  is  a  quantity  which, 
w^hen  substituted  for  the  unknown  quantity,  satisfies  the  equation. 
Thus,  3  and  1  are  the  roots  of  the  equation  x^  •—  4lX=:  —  3. 

183.  To  solve  an  Equation  is  to  find  its  roots. 

184.  A  Niimerical  Equation  is  one  in  which  all  the 
Tcnoion  quantities  are  represented  by  numbers.  Thus,  2a;2  -\-Zx  = 
V)x  +  15  is  a  numerical  equation. 

185.  A  Literal  Equation  is  one  in  w^hich  the  known 
quantities  are  represented  entirely  or  in  part  by  letters.  Thus, 
ax  -\-  b  ^=  ex  -\-  d  and  ax  —  5  =  3a;  —  5  are  literal  equations. 

186.  Hie  I>egree  of  an  equation  is  denoted  by  the  num- 
ber of  unknown  factors  in  that  term  which  contains  the  greatest 
number  of  such  factors.     Thus, 

ax  —  h=^c  is  of  the  first  degree, 
a:^  +  2/?a;  =zq  is  of  the  second  degree, 
a?y  -{-  x^  —  ex  z=z  a  is  of  the  third  degree, 
ic'*  +  «a;'*-i  -\-hx''~^  =  c  is  of  the  n^''  degi'ee. 

Remark. — It  should  be  observed  that  the  definition  implies  that  the 
equation  is  of  such  a  form  that  no  unknown  quantity  occurs  under  the 
radical  sign,  or  in  a  denominator. 

187.  A  Simple  Equation  is  one  of  the  first  degree. 

188.  A  Quadratic  Equation  is  one  of  the  second 
degree. 

189.  A  Cubic  Equation  is  one  of  the  third  degree. 

190.  A  biquadratic  Equation  is  one  of  the  fourth 
degree. 

191.  Higher  Equations  are  those  of  higher  degrees 
than  the  second. 


82  EQUATIONS. 

192.  For  brevity,  the  following  symbols  are  sometimes  used : 
.  • .  signifies  hence,  therefore,  or  consequently. 
• .  •  signifies  since,  or  because. 


TRANSFORMATION  OF  EQUATIONS. 

193.  To  Transform  an  equation  is  to  change  its  form 
without  destroying  the  equality  of  its  members. 

194.  Clearing  of  Fractions  and  Transposition 
of  Terms  are  the  principal  transformations. 

195.  To  clear  an  equation  of  fractions. 
Let  it  be  required  to  transform  the  equation, 

^l-Ve-' (^^' 

into  another,  all  of  whose  terms  shall  be  entire. 

Multiplying  both  members  of  (1)  by  al^c^  which  is  the  pro- 
duct of  the  denominators,  we  obtain 

lex  —  dbx  =  dtt^cd (2). 

Instead  of  multiplying  both  members  of  (1)  by  ab%  we  may 
clear  the  equation  of  fractions  by  multiplying  both  members  by 
dbc,  which  is  the  L.  C.  M.  of  the  denominators ;  we  thus  obtain 

ex  —  ax  =  abed (3). 


JR  ULE. 

Multiply  both  memhers  of  the  given  equation  by  the  product 
of  all  the  denominators  or  by  the  L.  C.  M.  of  all  the  denominators, 

196.  To  transpose  a  term  from  one  member  of  an 
equation  to  the  other. 

Let  us  consider  the  equation 

x-a  =  b-y    .     .    .    (1). 


TRAKSPOSITION.  83 

Adding  a  to  each  member  of  (1), 

x  —  a-\-a  =  b  —  y-{-a  (42,  2) ; 

that  is,  x=^d  -\-  a  —  y (2).. 

Subtracting  h  from  each  member  of  (2), 

x-lz=za-y (3)  (43,  3). 

Here  we  see  that  —  a  has  been  removed  from  one  member  of 
the  equation  and  appears  as  +  a  in  the  other ;  and  +  l  has  been 
removed  from  one  member  and  appears  as  —  ^  in  the  other. 

■Remove  the  term,  which  is  to  he  transposed,  from  the  member 
in  which  it  stands,  and  write  it,  with  its  sign  changed,  in  the 
other  member. 

Cor. — If  the  sign  of  every  term  in  an  equation  be  changed, 
the  equality  still  holds.    Thus,  if  x^a=b—y,  then  a—x=y—b* 


CHAPTEE   VII. 
SIMPLE    EQUATIOI^J'S 


SIMPLE  EQUATIONS  WITH  ONE  UNKNOWN   QUANTITY. 

197.  To  solve  a  simple  equation  containing  only- 
one  unknown  quantity. 

1.  Let  it  be  required  to  solve  the  equation 

3a;  _  4  =  24  —  X. 

By  transposition,      3a;  +  a;  =  24  +  4; 

that  is,  4a:  =  28 ; 

28 
whence,  by  division,  a;  =  —  =  7. 

We  may  verify  this  result  by  substituting  7  for  a;. in  the  given 
equation.  The  first  member  becomes  3x7  —  4,  that  is,  17; 
and  the  second  member  becomes  24  —  7,  that  is,  17. 

2.  Let  it  be  required  to  solve  the  equation 

^"T"  8  "^32' 

Multiplying  both  members  of  this  equation  by  96,  which  is  the 
L.  C.  M.  of  the  denominators, 

240a;  —  128a;  —  1248  =  60  +  3a;. 
By  transposition, 

240a;  —  128a;  —  3a;  =  1248  +  60 ; 
that  is,  109a;  =  1308; 

whence  by  division,  x  =  — — -  =  12. 

J  '  109 


ONE    UNKNOWl^    QUANTITY.  85 

RULE. 

I.  Clear  the  equation  of  fractions,  if  it  has  any,    ■ 

II.  Transpose  every  unhnown  term  of  the  second  meniber  to 
the  first,  and  every  known  term  of  the  first  member  to  the  second; 
and  reduce  each  member  to  its  simplest  form. 

III.  Divide  both  members  by  the  coefficient  of  the  unhnown 
quantity  ;  the  second  member  of  the  resulting  equation  will  be  the 
value  of  the  unhnown  quantity. 

IIjL  ustbations. 

Sometimes  it  is  more  convenient  to  clear  of  fractions  par- 
tially, and  then  to  effect  some  reductions  before  getting  rid  of  the 
remaining  fractional  coefficients.    For  example, 

1.  Solve 

x^l      22;  — 16       2a;  +  5_  ^,       3a;  +  7 

"n  3~"  +  "T~-^*  +  ~i2~- 

Multiplying  both  members  by  12, 
^^  ^^:^  '^^  -  4  (2a:  -  16)  +  3  (2a;  +  5)  =  64  +  32;  +  7; 

that  is,  ^^  (^  +  '^)  _  8:c  +  64  +  6a;  +  15  =  64  +  3a;  +  7. 


By  transposition  and  reduction, 

12  (a;  +  7)  _ 
11        "" 
Multiplying  by  11, 


5a; -8. 


12a;  +  84  =  55a;  — 88; 

by  transposition,      12a;  —  55a;  =  —  88  —  84; 

by  reduction,  —  43a;  —  —  172 ; 

by  changing  signs,  43a;  =  172 ; 

172 
by  division,  x  =  -j-^  =  4. 


86                                         SIMPLE    EQUATIONS. 

9     SoIto                                ^        _        ^ 

""                                 2x-\-l'~  5x  —  8' 

Multiplying  each  member  by  (2a;  +  1)  (ox  - 

-8), 

25a;  —  40  =  4a;  +  2. 

By  transposition  and  reduction, 

21a;  =  42; 

42 
by  division,                         a;  =  —  =  2. 

Verification, — Putting  this  value  for  x  in  the  given  equation, 
we  have 

or    1=1. 


3.  Solve 


4  +  1       10  —  8 

2a;  —  3       4a;  —  5 


3a;  —  4       6a;  —  7 
Clearing  of  fractions, 

(2a;  -  3)  {Qx  _  7)  =  (4a;  -  6)  (3a;  -  4) ; 

that  is,  12a;2  _  32a;  4-  21  =  12a?8  —  31a;  +  20. 

Subtracting  12a;2  fpom  ^Qth  members, 

21  — 32a;  =  20  — 31a;; 

by  transposition  and  reduction, 

—  a;  =  —  1,    or    a;  =  1. 

Verification. — Putting  this  value  for  z  in  the  given  equation, 

we  have 

2-3      4-5 


3-4~6-7' 


or    1=1. 


.    c!  1  a;       _       10a;       7 

4.  Solve  -_8  =  -3--3. 

Clearing  of  fractions, 

3a;  —  48  =  20a;  —  14. 

By  transposition  and  reduction, 

—  17a;  =  34,     or     17a;  =—34; 


OifE    UNKNOWN    QUANTITY.  87 

by  division,  x  =  —  —  =  —  2. 

2  20       7 

Verification,         -  —  8=     -^ —  ^>    or    — 9=—  9. 

5.  Solve  ax  -\-  b  =  ex  -\-  d. 

By  transposition,      ax  —  ex  =  d  —  b; 
that  is,  (a  —  c)  ic  =  (Z  —  6 ; 

d-b 


by  division. 


a— c 


Verifieation. — Putting  this  value  for  x  in  the  given  equation, 
we  have 

a  —  c                   a  —  c 
which  is  an  identity,  since  each  member  reduces  to . 

6.  Solve  Ix^iain  -\-  x). 

By  transposition,         bx  —  ax  =  an  ; 

by  division,  x  =  -. 

•^  b  —  a 

^    .^    ^.         ,         an  i  an  \  abn        abn 

Verincahon.    b  x  7 =^  ain  +  7 ),  or  t =7 . 

•^  b  —  a         \         b  —  aj         b  —  a     b  —  a 

198.  A  simple  equation  eontaining  only  one  unknoiun  quan- 
tity has  one  root,  and  no  more. 

By  clearing  of  fractions,  transposing,  and  reducing,  if  neces- 
sary, any  simple  equation  containing  only  one  unknown  quantity 
may  take  the  form  of 

ax-=b (1) ; 

whence,  x  =  - (2). 

a 

The  value  of  x  is  verified  thus : 

a  X  -  =  &,    or    b^b. 
a 


88  SIMPLE    EQUATION'S. 

Now,  it  is  evident  that  any  value  for  x  greater  than  -  would 

make  the  first  member  of  (1)  greater  than  the  second,  and  that 

any  value  for  x  less  than  -  would  make  the  first  member  of  (1) 

less  than  the  second.    Hence,  there  can  be  no  root  either  greater 

or  less  than  -;  that  is,  x  is  equal  to  -,  and  to  nothing  else. 

This  theorem  may  also  be  demonstrated  thus : 
Suppose  the  equation 

ax^=h (1), 

has  two  different  roots,  ^j  and  q ;  then  these  roots  will  satisfy 

equation  (1),  and  give 

ap  =  'b (2), 

aq  =  l) (3). 

Subtracting  (3)  from  (2), 

a(p-q)  =  0 (4)  (42,3); 

but  this  is  impossible,  for  a  is  not  zero,  and,  by  hypothesis,  p—  q 
is  not  zero. 

EXAMPLES. 

199.  Find  the  value  of  x  in  each  of  the  following  equations: 

.    22:  +  l       7z  +  5  . 

1.  — - —  =  — 5 — .  Ans,x  =  l, 

2.  I  _2  =  ^  +  ?-l.  Ans.  a:  =  20. 
•    2               4       0 

x  +  \       3^-4       l_6^  +  7 

3.  -y-+       ^       +g-       g      .  Arts.  X- 6. 

,    5a;— 11       ir— 1       11a;— 1  .  .. 

^     X         X         X         1  A  ^ 

^•2  +  3-4  =  3-  Ans.^^-^. 

6.  ^  +  -+^^16-^4^.  Ans.  ^  =  13. 

Z  o  4 


ONE    UNKNOWN     QUANTITY.  89 

11— a;       26  — a; 

7,  X  -\ —  =  — ^ — .  Ans,  x  =  S. 

1  ^5 

8.  19x  ^-{7x-2)=z4^  +  ^.  Ans.  x  =  l. 

_    ic— 3       a;— 4       a;— 5       x-\-l  . 


10.  — —  =  ^x  —  14.  Ans.  x  =  1. 


^  ^    x—S      2x  —  5       41       3a;  —  8      5a;  +  6 

11.-^ 6^=60  +  ^ 1^-    ^^''^  =  ^' 

12.  5£^_?:^:^5^_10.  ^^..  a;  =  3. 

13.  liS-x)  +a;-l|  =  -^-|.  Ans.  x  =  b. 

a;H-3      a;-2      3a;  —  5       1 
14.^ ^=-12- +4-  ^7^..a;  =  28. 

,„    3a; -1       13— a;       7a;       11  (a;  +  3) 

._    5a;  — 3       9— a;       5a;       19,         ,, 

10.  — ^ ^  z=  —  -f  —  (a;  —  4).  ^W5.  a;  =  2. 

.„    5a;  — 1       9a;  — 5       9a;  —  7 

17.  — jt; 1 -j —  =  — - — .  Ans.  a;  =  3. 

7  11  0 

^„   3a;  +  5       2a;  +  7       ,^      3a;      ^ 

18.  — Y ^  +  10  -  y  =  0.  Ans.  X  =  10. 

a;-l       4a; -f       7a;  -  6  _  ^   ,  a;-2  .   3a;  -  9      » 

^/i.<f.  x  =  —. 

20.  (x+|)(x-|)-(x+5)(a;-3)+|  =  0. 

^7i5.  a;  =:  12. 


''■\{^-t}-\{^-lh\(^-t}='-   ^-- 


25* 


9D  SIMPLE    EQUATIONS. 

22.  (a+x)  {b-{-x)  =  (c  +  x)  (d-^x).       Ans,  x  =  ^^^~^_^ 

23.  ^  +  -^  =  -±-,  Ans.  X  =  ^r^^. 
a      b—a      b-\-a  b{o  -{-  a) 

%Lax  +  b  =  -^+-^.  Ans.x  =  ^^^^. 

^     x—a       x—b      x—c      X—  {a  -[•  b  +  c) 
b  c  a  abc 

a?c  +  ab^  -\-  bc^  —  a  —  b  —  c 


Ans.  X  = 


ac  -\-  be  -\-  ab  —  1 


26.  (a  +  x)  {b  -^  x)  -  a{b  +  c)  =^  +  x?.        Ans.  x=j, 

c^^   a-\-b         a  b  '     .  ab(a-\-b—2c) 

27.  = 1 T.  Ans.  x  = 


x—c      x—a      x—b'  '  a^-\-b^—ac—bc 

^^   ax^  -\-  bx  -^  c      ax  +  b  .  hr  —  cq 

28. — 5 ' —  = .  Ans.  x  = *. 

px^  -\-  qx  -\-  r      px  -{■  q  cp  —  ar 

a-\-b      (a-\-bY        a{a-\-by  a 

.  db 

Ans.  x.= 


a-\-h' 

^^  mix-{-a)       n{x-[-b)  .  bn  —  am 

30.  — ^^ — -r^  +  -^^ — — ^  =  m  +  n,  Ans.  x  = . 

x-i-b  x-{-a  m  —  n 

/x  —  a\^     x  —  2a  —  b  ^  a  —  b 

ol. 


/x  —  ay      x  —  2a  —  b  . 

L.  I 7 )  = rr .  Ans.  X  = 

\x  -\-  bl       X  -\-  a  +  2b 


2 


32.  (x  —  ay-{-  (x  —  by-\-  (x  —  cy  =  S  {x  —  a){x  —  b)  {x  —  c) 

•  -                                                         .  a  +  b  +  c 

Ans.  X  = . 

33.  .15a:  +  1.575  —  .875a;  =  .0625a;.  Ans.  x  =  2. 

34.  1.2a;  -  '^^^  7  '^^  =  .4a:  H-  8.9.  Ans.  x  =  20. 

.5 

70^ OK 

35.  4.8a:  -          ,         =  1.6a;  +  8.9.  Ans.  x  =  6. 


ONE    UNKNOWN"   QUANTITY.  91 

SOLUTION    OF    PROBLEMS. 

300.  We  shall  now  apply  the  methods  already  given  to  the 
solution  of  some  problems,  and  thus  exhibit  to  the  student  speci- 
mens of  the  use  of  Algebra. 

In  a  problem  certain  quantities  are  given,  and  certain  others, 
which  have  some  assigned  relations  to  them,  are  to  be  found. 

301.  The  solution  of  a  problem  by  Algebra  consists  of  twc 
distinct  parts : 

1st.  The  Statement — that  is,  the  formation  of  the  equation 
which  shall  express  the  relation  between  the  known  and  the  un- 
known quantities. 

2d.  The  Solution  of  the  equation. 

303.  Sometimes  the  conditions  of  the  problem  are  such  as  to 
famish  the  equation  directly  ;  and  sometimes  it  is  necessary,  from 
the  given  conditions,  to  deduce  others,  from  which  to  form  the 
equation.  When  the  conditions  furnish  the  equation  directly, 
they  are  called  Explicit  Conditions.  Wlien  the  conditions  are 
deduced  from  those  given  in  the  problem,  they  are  called  Implicit 
Conditions. 

303.  It  is  impossible  to  give  any  precise  rule  for  solving 
every  problem;  the  following  directions,  however,  may  furnish 
some  aid : 

Denote  the  unhiown  quantity  by  one  of  the  final  letters  of  the 
alphabet,  and  express,  in  algebraic  language,  the  relation  which 
subsists  between  the  unknotvn  quantity  and  the  given  quantities  ; 
an  equation  loill  thus  be  obtained  from  which  the  value  of  the  un- 
known quantity  may  be  found, 

IIjJj  USTRATION8, 

1.  The  sum  of  two  numbers  is  89,  and  their  difference  is  31 ; 
find  the  numbers. 

Let  X  denote  the  less  number,  then  the  greater  number  will 
be  31  -h  a; ;  .  • .  since  their  sum  is  89, 

31  +  a;  +  a:  =  89; 
that  is,  31  +  2a;  =  89. 


92  SIMPLE    EQUATIONS. 

By  transposition,    2a;  =  89  —  31  =  58 ; 
whence,  x  =  -^  =  29,  the  less  number. 

.*.  the  greater  number  is  29  +  31,  that  is,  60. 

2.  A  bankrupt  owes  B  twice  as  much  as  he  owes  A,  and  C  as 
much  as  he  owes  A  and  B  together ;  out  of  $300  which  is  to  be 
divided  among  them,  what  should  each  receive  ? 

Let  X  denote  the  number  of  dollars  which  A  should  receive ; 
then,  by  the  conditions  of  the  problem,  2x  will  be  the  number  of 
dollars  B  should  receive,  and  x  +  2x,  that  is,  3a;,  will  be  the  num- 
ber of  dollars  0  should  receive.    They  together  receive  $300 ; 

.*.    a;  4- 2a;  +  3a;  =  300. 

Reducing,  6a;  =  300; 

whence,  x  =  — r-  =  50 ; 

b 

therefore,  A  should  receive  $50,  B  $100,  and  C  $150. 

3.  Divide  a  line  21  inches  long  into  two  parts,  such  that  one 
may  be  three-fourths  of  the  other. 

3a; 
Let  X  denote  the  number  of  inches  in  one  part,  then   — 

will  denote  the  number  of  inches  in  the  other  part ;  .*.  (42,  1), 

.  +  5  =  21. 

Clearing  this  equation  of  fractions, 
4a;  +  3a;  =  84; 
by  reduction,  7a;  =  84; 

whence,  a;  =  -^^  =  12 ; 

therefore,  one  part  is  12  inches  long,  and  the  other  part  9  inches. 

4.  If  A  can  perform  a  piece  of  work  in  8  days,  and  B  in  10 
days,  in  what  time  will  they  perform  it  together  ? 

Let  X  denote  the  number  of  days  required. 


ONE    UNKKOWK    QUANTITY.  93 

In  one  day  A  can  perform  ^th  of  the  work,  therefore  in  o> 

days  he  can  perform  -  ths  of  the  work. 

In  one  day  B  can  perform  T^rth  of  the  work,  therefore  in  x 

days  he  can  perform  —  ths  of  the  work.    Hence,  since  A  and  B 
together  perform  the  whole  work  in  x  days, 

8^  10 
By  clearing  of  fractions, 

6a;  +  4a:  =  40 ; 

by  reduction,  Oa-  =  40 ; 

whence,  x  =  -^  =  4^, 

5.  A  laborer  was  employed  for  20  days,  on  condition  that  for 
every  day  he  worked  he  should  receive  50  cents,  and  for  every 
day  he  was  idle  he  should  forfeit  25  cents.  At  the  end  of  the  20 
days  he  received  $4 ;  how  many  days  did  he  work,  and  how  many 
days  was  he  idle  ? 

Let  X  denote  the  number  of  days  he  worked ;  then  he  was  idle 

20  —X  days. 

50a;  =  wages  due  for  work,  and 

25  ^0.—  x)  =  the  amount  he  forfeited ; 
.-.  50a;  — 25(20  — rr)  =  400; 
that  is,  50a;  —  500  +  25a;  =  400 ; 

by  transposition  and  reduction, 

75a;  =  900 ; 
whence,  a;  =  12  =  the  number  of  days  he  worked, 

and  20  —  a;  =  8  =  the  number  of  days  he  was  idle. 


94  SIMPLE    EQUATIONS. 

6.  How  much  rye,  at  54  cents  a  bushel,  must  be  mixed  with 
50  bushels  of  wheat,  at  72  cents  a  bushel,  iu  order  that  the  mix- 
ture may  be  worth  60  cents  a  bushel  ? 

Let  X  denote  the  number  of  bushels  required ;  then  h\x  is  the 
value  of  the  lye ;  and  since  the  value  of  the  wheat  is  3600,  the 
value  of  the  mixture  is  54a;  +  3600. 

The  value  of  the  mixture  is  also  (x  +  50)  60  \ 

(a;  +  50)  60  =  54a;  +  3600; 
that  is,  60a;  +  3000  =  54a;  +  3600 ; 

by  transposition  and  reduction, 

^x  =  600 ; 
whence,  x  =  100. 

7.  A  smuggler  had  a  quantity  of  brandy,  which  he  expected 
would  sell  for  ^100 ;  after  he  had  sold  10  gallons,  a  revenue  officer 
seized  one- third  of  the  remainder,  in  consequence  of  which,  the 
smuggler  received  only  $80  for  his  brandy.  How  many  gallons 
had  he  at  fii-st,  and  what  was  the  price  per  gallon  ? 

Let  X  =  the  number  of  gallons ;   then  —  is  the  price  per 

X 

X 10 

gallon,  and  — r —  is  the  quantity  seized,  the  value  of  which  is 
o 

100  —  80  =  20.     The  value  of  the  quantity  seized  is  also  ex- 

^  ^    .-c  —  10      100 
pressed  by  — ^ —  x 


X 


l«xi^  =  20. 


3  a; 

Clearing  of  fractions, 

100  (a; -10)  =60a;; 
that  is,  100a;— 1000=60a;; 

by  transposition  and  reduction, 

40a;  =  1000; 

whence,  x  =  25,  the  number  of  gallons ; 

,  100      100      ,  -,  11       ..        .  11 

and  —  =  --^  =  4  dollars,  the  price  per  gallon. 


OKE    UN^Kiq^OWK    QUANTITY.  95 

204c,  rMOBJLEMS. 

1.  Divide  13870  between  two  persons,  A  and  B,  so  that  A 
eliall  receive  twice  as  mucli  as  B. 

Ans.  A  gets  12580,  and  B  $1290. 

2.  Divide  $420  between  A  and  B,  so  that,  for  every  dollar  A  re- 
ceives, B  may  receive  $2 J.    Ans.  A's  share  =  $120,  B's  =  1300. 

3.  How  much  money  is  there  in  a  purse  when  the  fourth  part 
and  the  fifth  part  together  amount  to  145  ?  A7is.  $100. 

4.  After  paying  the  seventh  part  of  a  bill  and  the  fifth  part, 
$92  were  still  due ;  what  was  the  amount  of  the  bill  ? 

Ans.  $140. 

5.  Divide  46  into  two  parts,  such  that  if  one  part  be  divided 
by  7  and  the  other  by  3,  the  sura  of  the  quotients  shall  be  10. 

Ans.  28  and  18. 

6.  A  company  of  266  persons  consists  of  men,  women,  and 
children;  there  are  four  times  as  many  men  as  children,  and 
twice  as  many  women  as  children.     How  many  of  each  are  there  ? 

Ans.  38  children,  76  women,  152  men. 

7.  A  person  expends  one-third  of  his  income  in  board  and 
lodging,  one-eighth  in  clothing,  one-tenth  in  charity,  and  saves 
$318.    What  is  his  income  ?  Ans.  %^%^. 

8.  Three  towns.  A,  B,  C,  raise  a  sum  of  $594 ;  for  every  dollar 
B  contributes,  A  contributes  three-fifths  of  a  dollar,  and  C  seven- 
eighths  of  a  dollar.    What  does  each  contribute  ? 

Ans.  A  contributes  $144,  B  $240,  C  $210. 

9.  Divide  $1520  among  A,  B,  and  C,  so  that  B  shall  have  $100 
more  than  A,  and  C  $270  more  than  B. 

Ans.  A  gets  $350,  B  $450,  C  $720. 

10.  A  certain  sum  of  money  is  to  be  divided  among  A,  B,  and 
C.  A  is  to  have  $30  less  than  the  half,  B  is  to  have  $10  less  than 
the  third  part,  and  C  is  to  have  $8  more  than  the  fourth  part. 
What  does  each  receive?  Ans.  A  $162,  B  $118,  C  $104. 

11.  The  sum  of  two  numbers  is  5760,  and  their  difference  is 
equal  to  one-third  of  the  greater;  find  the  numbers. 

Ans,  3456  and  2304. 


96  SIMPLE    EQUATIONS. 

12.  Two  casks  contain  equal  quantities  of  beer ;  from  the  first 
34  quarts  are  di*awn,  and  from  the  secon(J  80 ;  the  quantity  re- 
maininir  in  tiie  first  cask  is  now  twice  that  in  the  second.  How 
much  did  each  cask  originally  contain  ?  Ans.  126  quarts. 

13.  A  person  bought  a  picture  at  a  certain  price,  and  paid  the 
same  price  for  a  frame ;  if  the  frame  had  cost  $1  less,  and  the  pic- 
ture $J  more,  the  price  of  the  fi-ame  would  have  been  only  half 
that  of  the  picture.    What  was  the  price  of  the  picture  ?  ^ 

Ans.  $2}. 

14.  A  house  and  garden  cost  $850,  and  five  times  the  price  of 
the  house  was  equal  to  twelve  times  the  price  of  the  garden ;  find 
the  price  of  each.     •  A71S,  House,  $600 ;  garden,  $250. 

15.  One-tenth  of  a  rod  is  colored  red,  one- twentieth  orange, 
one-thirtieth  yellow,  one-fortieth  green,  one-fiftieth  blue,  one- 
sixtieth  indigo,  and  the  remainder,  which  is  302  inches  long, 
violet;  find  the  length  of  the  rod.  Ans.  400  inches. 

16.  Two-thirds  of  a  certain  number  of  persons  received  $18 
each,  and  one-third  received  $30  each.  The  whole  sura  received 
was  $660.    How  many  persons  were  there  ?  A7is.  30. 

17.  Find  the  number  whose  tliird  paii;  added  to  its  seventh 
part  gives  a  sum  equal  to  20.  Ans.  42. 

18.  The  difference  between  the  squares  of  two-  consecutive 
numbers  is  15.     What  are  the  numbers  ?  Ans.  7  and  8. 

19.  A  performs  ^  of  a  piece  of  work  in  4  days ;  he  then  re- 
ceives the  assistance  of  B,  and  the  two  together  finish  it  in  6 
days.    Find  the  time  in  which  each  alone  can  do  the  whole  work. 

A71S.  A  in  14  days;  B  in  21  days. 

20.  A  bought  eggs  at  18  cents  a  dozen ;  had  he  bought  5  more 
for  the  same  money,  they  would  have  cost  him  2^  cents  a  dozen 
less.     How  many  eggs  did  he  buy  ?  Atis.  31. 

21.  A  man  bought  a  certain  number  of  sheep  for  $94 ;  having 
lost  7  of  them,  he  sold  one-fourth  of  the  remainder  at  prime  cost 
for  $20.     How  many  sheep  did  he  buy?  Ans.  47. 

22.  A  man  leaves  home  in  a  stage  which  travels  12  miles  an 
hour,  and  agrees  to  return  in  2  hours.  How  far  may  he  ride  if  he 
walks  back  at  the  rate  of  4  miles  an  hour?  Ans.  Q  miles. 


OIJE    UNKNOWN    QUANTITY.  97 

23.  A  and  B  play  at  a  game,  agreeing  that  the  loser  shall 
always  pay  to  the  winner  $1  more  than  half  the  money  the  loser 
has;  they  commence  with  equal  sums  of  money,  hut  after  B  has 
lost  the  first  game  and  won  the  second,  he  has  twice  as  much  as 
A;  how  much  had  each  at  the  beginning?  Ans.  $6. 

24.  A  person  who  possesses  $12000  uses  a  portion  of  the 
money  in  building  a  house.  One-third  of  the  money  which  re- 
mains he  invests  at  4  per  cent.,  and  the  other  two- thirds  at  5  per 
cent.,  and  from  these  investments  he  obtains  an  income  of  $392. 
What  was  the  cost  of  the  house  ?  A7is.  $3600. 

25.  A  takes  from  a  purse  $2  and  one-sixth  of  what  remains ; 
then  B  takes  $3  and  one-sixth  of  what  remains ;  they  then  find 
that  they  have  taken  equal  amounts.  How  many  dollars  were  in 
the  purse,  and  how  many  did  each  take  ? 

Ans.  There  were  $20  in  the  purse,  and  each  took  $5. 

26.  A  vessel  can  be  emptied  by  three  taps ;  by  the  first  alone 
it  could  be  emptied  in  80  minutes ;  by  the  second  alone,  in  200 
minutes ;  and  by  the  third  alone,  in  5  hours.  In  what  time  will 
the  vessel  be  emptied  if  all  the  taps  are  opened?     Ans.  48  min. 

27.  A  person  buys  some  tea  at  36  cents  a  pound,  and  some  at 
60  cents  a  pound ;  he  wishes  to  mix  them,  so  that,  by  selling  the 
mixture  at  44  cents  a  pound,  he  may  gain  10  per  cent,  on  each 
pound  sold ;  find  how  many  pounds  of  the  inferior  tea  he  must 
mix  with  each  pound  of  the  superior.  A?is.  5. 

28.  A  cask,  A,  contains  12  gallons  of  wine  and  18  gallons  of 
water ;  another  cask,  B,  contains  9  gallons  of  wine  and  3  gallons 
of  water ;  how  many  gallons  must  be  drawn  from  each  cask,  so  as 
to  produce,  by  their  mixture,  7  gallons  of  wine  and  7  gallons  of 
water?  Ans.  10  from  A,  and  4  from  B. 

29.  A  can  dig  a  ditch  in  one-half  the  time  that  B  can ;  B  can 
dig  it  in  two-thirds  of  the  time  that  C  can ;  all  together  they  can 
dig  it  in  6  days ;  find  the  time  in  which  each  alone  can  dig  the 
ditch.  Ans.  A  in  11  days,  B  in  22  days,  and  C  in  33  days. 

30.  At  what  time  between  one  o'clock  and  two  o'clock  is  the 
minute  hand  exactly  one  minute  in  advance  of  the  hour  hand? 

Ans.  6-j^  minutes  past  one. 
7 


98  SIMPLE    EQUATIONS. 

31.  A  man  leaves  home  in  a  stage  which  travels  h  miles  an 
hour,  and  agrees  to  return  in  a  hours.  How  far  may  he  ride,  if 
he  walks  back  at  the  rate  of  c  miles  an  hour  ? 

Ans.  -r miles. 

b-\-c 

ScH. — As  «,  h,  and  c  may  have  any  values  whatever,  the  solu- 
tion of  Problem  31  furnishes  a  formula  which  can  be  used  for  the 
solution  of  any  similar  problem.  Thus,  to  obtain  the  answer  to 
Problem  22,  we  have  only  to  substitute  2  for  a,  12  for  Z>,  and  4 
for  c,  which  gives 

2  X  12  X  4      96       _ 

^=-T2-+-r-  =  i6  =  ^- 

A  problem  is  said  to  be  generalized  when  letters  are  used  to 
represent  its  known  quantities. 

32.  A  crew,  which  can  row  a  boat  at  the  rate  of  9  miles  an 
hour  in  still  water,  finds  that  it  takes  twice  as  long  to  come  up  a 
river  as  to  go  down ;  at  what  i-ate  does  the  river  flow  ? 

Ans,  3  miles  an  hour. 

33.  A  certain  article  of  consumption  is  subject  to  a  duty  of 
72  cents  per  cwt. ;  in  consequence  of  a  reduction  in  the  duty,  the 
consumption  increases  one-half,  but  the  revenue  falls  one-third. 
Find  the  duty  per  cwt.  after  the  reduction. 

Ans.  32  cents  per  cwt. 

34.  A  merchant  maintained  himself  for  3  years  at  a  cost  of 
^250  a  year ;  and  in  each  of  those  years  augmented  that  part 
of  his  stock  which  was  not  so  expended  by  one-third  thereof.  At 
the  end  of  the  third  year  his  original  stock  was  doubled ;  what 
was  that  stock  ?  Ans.  $3700. 

35.  A  market  woman  bought  some  eggs  at  2  for  a  cent,  and  as 
many  more  at  3  for  a  cent ;  she  sold  them  all  at  the  rate  of  5  for 
2  cents,  and  found  she  had  lost  4  cents.  How  many  did  she  buy 
ofea<jhsort?  Ans.  120. 

36.  A  man  hired  a  servant  for  one  year  at  the  wages  of  ^90 
and  a  suit  of  clothes.  At  the  end  of  7  months  the  servant  quits 
his  service  and  receives  833.75  and  the  suit  of  clothes.  At  what 
price  were  the  clothes  estimated  ?  Ans.  Ho. 


TWO    UNKNOWN    QUANTITIES.  99 

37.  A  general  arranging  his  men  in  the  form  of  a  sohd  square, 
finds  he  has  21  men  over ;  but  attempting  to  add  one  man  to 
each  side  of  the  square,  finds  he  wants  200  men  to  fill  up  the 
square;  find  the  number  of  men.  Ans.  12121. 

38.  A  boatman  can  row  14  miles  an  hour  with  the  tide; 
against  a  tide  two-thirds  as  strong,  he  can  row  only  four  miles  an 
hour.     What  is  the  velocity  of  the  tide  in  each  case  ? 

Ans.  6  miles,  and  4  miles. 

39.  Two  men  start  from  the  same  point  fit  the  same  time,  and 
travel  in  the  same  direction ;  the  first  steps  twice  as  far  as  the 
second,  but  the  second  makes  five  steps  while  the  first  makes  one. 
At  the  end  of  a  certain  time  they  are  300  feet  apart ;  how  far  has 
each  traveled?  Ajis.  1st,  200  feet;  2d,  500  feet. 

40.  A  ship  sails  w4th  a  supply  of  biscuit  for  60  days,  at  a  daily 
allowance  of  a  pound  a  head ;  after  being  at  sea  20  days  she  en- 
counters a  storm,  in  which  5  men  are  washed  overboard,  and  dam- 
age sustained  that  will  cause  a  delay  of  24  days,  and  it  is  found 
that  each  man's  daily  allowance  must  be  reduced  to  five-sevenths 
of  a  pound.    Find  the  original  number  of  the  crew.     A7is.  40. 

SIMPLE  EQUATIONS  WITH  TWO  UNKNOWN  QUANTITIES. 

205.  Suppose  we  have  an  equation  containing  two  unknown 
quantities,  a:  and  «/;  for  example, 

6x-2y  =  4:    .     .     .    .     (1). 

For  every  value  which  we  please  to  ascribe  to  one  of  the  un- 
known quantities  we  can  determine  the  corresponding  value  of 
the  other,  and  thus  find  as  many  pairs  of  values  as  we  please. 

r  8 

Thus,  if  w  =  1,  we  find  x  =  j;    if  ?/  =  2,  we  find  x  =  -;    and 

so  on. 

Suppose  we  have  another  equation  of  the  same  kind;   for 

example, 

4:^  +  3^  =  17      .    .     .     (2). 

We  can  also  find  as  many  pairs  of  values  as  we  please  which 
satisfy  this  equation. 


100  SIMPLE    EQUATIONS. 

But  suppose  we  ask  for  values  of  x  and  y  which  satisfy  both 

equations;  we  shall  find  then  that  there  is  only  one  vakie  of  x 

and  one  value  of  y.    For,  multiplying  (1)  by  3,  and  (2)  by  2,  we 

have 

lbx  —  (jy  =  12    .     .    .     (3),  . 

8^  +  6y  =  34     ...     (4). 
Adding  (3)  and  (4),  member  to  member  (43,  2), 

23a;  =  46 (5); 

whence,  a:  =  2. 

We  may  now  find  the  value  of  y  by  substituting  2  for  x  in 
either  of  the  given  equations.     Substituting  2  for  x  in  (1), 

10  — 22^  =  4   .    .     .     .     (6); 
whence,  ^  =  3. 

Hence,  if  hoth  equations  are  to  be  satisfied,  x  must  be  equal  to 
2,  and  y  must  be  equal  to  3. 

206.  Siimiltaneous  Equations  are  two  or  more  equa- 
tions which  are  to  be  satisfied  by  the  same  values  of  the  unknown 
quantities.  We  are  now  about  to  treat  of  simultaneous  equations 
of  the  first  degree  involving  two  unknown  quantities. 

307.  There  are  three  methods  which  are  usually  given  for 
solving  these  equations.  The  object  of  each  method  is  to  obtain 
from  the  two  given  equations  a  single  equation  containing  only 
one  of  the  unknown  quantities.  The  unknown  quantity  which 
does  not  appear  in  the  resulting  equation  is  said  to  be  eliminated. 

208.      ELIMINATION  BY  ADDITION  OR  SUBTRACTION. 

1.  Let  it  be  required  to  solve  the  equations, 

4:X  +  3y  =  22 (1), 

5x-':y  =  6 (2). 

Multiplymg  (1)  by  7,  and  (2)  by  3, 

28:c  +  217/ =  154 (3), 

15x- 21^^  =  18 (4). 


'^(/Wcu^^^pr^^, 


TWO    UKKXOWX    QUANTITIES.  101 


Adding  (3)  and  (4), 

43:^  =  172 

(5); 

whence, 

a;  =  4. 

Substituting 

4  for  X  in  (1), 

IG  +  3^  =  22; 

whence, 

y  =  % 

2.  Let  it  be 

required  to  solve  the  equations. 

22:  +  7?/  =  29.     .    .     . 

(1), 

32:  +  5?/  =:  27      ... 

(2). 

Multiplying  (1)  by  5,  and  (2)  by  7, 

mr  4- 35?/ =  145     .     . 

(3), 

21a;  +  35?/  =  189     .    . 

(4). 

Subtracting  (3)  from  (4), 

. 

nx  =  u 

(5); 

whence, 

«  =  4. 

Substituting 

;  4  for  a;  in  (1), 

8  +  7^  =  29 ; 

whence, 

y  =  3. 

RULE. 

I.  Multiply  or  divide  the  given  equations  by  such  quantities 
that  the  coefficients  of  the  quantity  to  he  eliminated  shall  he  equal 
in  the  two  resulting  equations. 

II.  If  these  equal  coefficients  have  like  signs,  subtract  one  of 
the  resulting  equations  from  the  other,  member  from  member ;  if 
they  have  unlike  signs,  add  the  equations,  member  to  member. 

ScH.  1.— Before  commencing  the  operation  of  elimination, 
each  of  the  given  equations  should  be  reduced  to  the  form  of 
ax  +  by  =  c,  if  it  is  not  already  of  that  form. 


102  SIMPLE    EQUATIONS. 

ScH.  2. — The  coefficients  of  the  quantity  to  be  ehtninated 
may  be  equal  in  the  given  equations ;  in  that  case,  the  first  step 
in  the  rule  is  unnecessary. 

ScH.  3. — In  preparing  the  given  equations  by  multiplication, 
it  is  best  to  divide  the  L.  C.  M.  of  the  coefficients  of  the  quantity 
to  be  eliminated  by  each  of  these  coefficients ;  the  quotients  thus 
obtained  ^vvill  be  the  least  multipliers  that  can  be  used. 

ScH.  4. — It  is  genemlly  convenient  to  clear  the  equations  of 
fractions,  if  they  have  any,  before  applying  the  rule.  This  is  not 
necessary,  however.  For  if  the  quantity  to  be  eliminated  has 
fractional  coefficients  in  the  two  equations,  they  may  be  reduced 
to  equivalent  fractions  having  a  common  denominator;  it  will 
then  be  necessary  to  render  the  numerators  equal  by  multiplica- 
tion or  division,  according  to  the  rule. 

EXAMPLES. 

Solve  the  following  groups  of  simultaneous  equations : 


(7a:  — 41/  =  19) 

I  3a; +  4// =  38) 

j    a;  +  3y  =  10) 
\dx-\-^=    9) 

4    \  ^-\(y-  2)  =  5) 

•    (4z/-i(a:  +  10)  =  3f 
\x-\-y  =  s  ) 
ix  —  y  =  d) 


5 

201>«  ELIMINATION    BY    SUBSTITUTION. 

Let  it  be  required  to  solve  the  equations 

Ax-\-3ij  =  22      ...     (1), 
bx  —  7y  =  Q.    .    .    .     (2). 

From  (1)  we  find 

22  -  4a;  ,„. 

y  =  — o —  ....    (3). 


Ans, 

x  = 

5, 

y  = 

4. 

Ans. 

X  — 

G, 

y  = 

5. 

Ans. 

X  = 

1, 

1/  = 

3. 

Ans, 

x  = 

5, 

y  = 

2. 

Ans. 

X  = 

.S' 

-\-d 
2     ' 

s 

2 

d 

TWO    UNKNOWN    QUANTITIES. 


103 


Substituting  this  value  for  y  in  (2), 

22  — 4a; 


that  is, 


bx 


5a;- 7  X 
154  -  28a; 


3 
=  6  . 


=  6; 
.     (4). 


Multiplying  (4)  by  3, 

15a;  —  154  4- 28a;  =  18  .    .    (5); 
by  transposition  and  reduction, 

43a;  =  172; 
whence,  a;  =  4. 

Substituting  4  for  x  in  (3),  we  find  y  =  2. 

RULE. 

Find,  from  one  of  the  give?i  equations,  an  expression  for  the 
value  of  the  unknown  quantity  to  be  eliminated,  and  substitute 
this  value  for  the  same  unknown  quantity  in  the  other  equation  ; 
there  loill  thus  be  formed  a  iiew  equation  containing  only  one  un- 
knowti  quantity. 

EXAMPLES. 

Solve  the  following  groups  of  simultaneous  equations : 


1. 


j  2a;  +  3?/  =  33  ) 
(  4a;  +  5?/  =  59  ) 

x-{-y      x—y_ 

2  3     ~ 

x  +  y       x—y 

3  "^     4 


11 


3. 


(3a:-2^=:l) 
(  3?/  -  4a;  =  1  f 


4. 


2^3 


^  4.  ^ 

3'^4 


Ans.  x=i^,  «/  =  7. 


Ans.  a;  =  18,  ?/  =  6. 


Ans,  x  =  6,  y  =  'il' 


Ans.  a?  =  —  6,  y  =  12, 


104  SIMPLE    EQUATIONS. 


5. 


X      y 

a      b  !       ^    ,      _  (ic{dn—hm)      _  hd{am—cn) 

X      y  '     ~     ad  —  be    ^  ^  ~      ad— be 

'c'^d^ 


210.  ELIMINATION   BY   COMPARISON. 

Let  it  be  required  to  solve  the  equations 

72: +  62^  =  20 (1), 

9a; —  4^  =  14 (2). 

From(l),  y  =  ^^^ (3), 

g-P 14 

and  from  (2),  y  = j (4) ; 

9^—1^      20  —  72;  ,_. 

.-.(42,6),  ___-  =  ___    .    .    .    (5). 

Clearing  of  fractions, 

27a;  — 42  =  40  — 14ar; 
whence,  a;  =  2. 

Substituting  2  for  x  in  either  (3)  or  (4),  we  find  y  =  1. 

Br  ULE. 

Find,  from  each  of  the  given  equations,  an  expression  for  the 
vahte  of  the  unknown  quantity  to  be  eliminated,  and  equate  the 
expressions  thus  obtained ;  an  equation  will  thus  be  formed  con- 
taining only  one  unknown  quantity. 

EXAMTIjES. 

Solve  the  following  groups  of  simultaneous  equations : 

^     j4a;  — 2v=    20)  ,  _  ^- 


TWO    Ui^'KITOWN    ^UAifTITIES. 


105 


(2a;—   v=    1) 

I 


4. 


5. 


1/ 

S  7a;  -  3?/  =  12 
(  2a;  +  2i/  =  12 

a:+    y  =  21 


^x  -  y      3_3^ 

-1 2-T~'~^ 


a;  +  2/ 


2| 


Ans.  X  =z2,  y  =  3. 
Ans.  x  =  3,  y  =  3. 

Ans.  x  =  9,  y  =  12, 
Ans.  a;  =  3,  y  =  5. 


211,  Genekal  Scholium. — In  the  solution  of  simultaneous 
equations,  any  of  the  preceding  methods  of  ehmination  can  be 
used,  as  may  be  most  convenient,  each  method  having  its  advan- 
tages in  particular  cases.  Generally,  however,  the  equation  ob- 
tained by  using  the  second  or  third  method  contains  fractional 
terms.  This  inconvenience  is  avoided  if  we  eliminate  by  the  first 
method.  The  second  method  may  be  preferable  whenever  the  co- 
eflBcient  of  one  of  the  unknown  quantities  in  one  of  the  given 
equations  is  unity;  for  then  the  inconvenience  of  which  we  have 
just  spoken  may  be  avoided.  We  shall  sometimes  have  occasion 
to  use  the  second  and  third  methods,  but  generally  the  first 
method  is  preferable. 


EX  A  MP  JjES 


Solve  the  following  groups  of  simultaneous  equations : 

1.  i^  +  2/  =  in.  ^^,.^^^11, 2/ =4. 

[x  —  y=l)  ^ 

(3a;-2y  =  l) 
\3y-^x  =  l) 


(3a;-5y  =  13) 
(  2a;  +  7y  =  81  f 

(    2a;-f-3y  =  43) 
*    (lOa;—    y=    7) 


Ans.  a;  =  5,  «/  =  '^* 
Ans.  a;  =  16,  y  =  Z 
Ans.  x  =  2,  y  =  l3. 


106 


SIMPLE    EQUATIONS. 


(    6x-    1y=    33) 
( 11a;  +  127/  =  100  ) 


i3^-7.=    4) 

j  21y  +  20^  =  165  ) 
=  295) 


(  77y  _  30a; 

(    5a: +  7?/ =  43) 
( 11a;  +  9«/  =  69  r 

(8a;-21y=    33) 
(6a; +  352/ =  177) 

(    5a;  -f-    7y  =  41  ) 

j  16a;  +  17y  =  500  ) 
(17a:-    3?/ =  110) 


12. 


13.   \ 


14. 


15.   J 


5  +  6  -  ^^ 


^-^-21 
2       4~ 


-  +  ^-9 
3  +  4-*^ 

?  +  ^=7 
4^6 


2       3~ 

^  +  ^  =  1 
3^4 


a:  +  i2/       x—y        ^ 
"2 3"=    ^ 

~3  4"-    ^ 


^W5.  a:  =  8,  y  =  1. 
Ans.  x=z2,  y  =  6. 
Ans.  a:  =  3,  y  =  5. 
Ans.  a:  =  3,  ?/  =  4. 
^W5.  x  =  12,  y  =  3. 
^?J5.  a;  =  4,  y  =  d' 
Ans,  a:  =  10,  ?/  =  2ft 

^W5.  a;  =  60,  ^  =  36i 
Ans.  X  =  12,  ?/  =  20. 
Ans.  x=  —6,  y=Vl 


Ans.  a:  =  18,  y  =  Q. 


TWO    UNKNOWN    QUANTITIES. 


16. 


17. 


18. 


19. 


11a;  —  oy_^x-\-y 

n        "      16 
8a;  — 5^=1 


3        4  +  2+^-^       4+12 
|_|  +  2=^-2.  +  6 


x  =  4.y 

\{%x  +  ly)^l=.h2x-Qy-\-l) 


107 

Ans,  X  =  11,  y  =zll, 

Ans.  x  =  2f  y='7. 
Ans.  a;=4,  y=l. 


x  +  l(3x-y-l)^l  +  l{y-l) 


20. 


21. 


5(4x  +  3y)=g  +  2 


'3a;— -Sy  ,   ,      2x  +  y 

x  —  2y_x      y 
^ ^-2  +  3 

3aj       ;/        4  _  a;         «/ 
10  ~"  15  ~  9  ~  12  ~  18 

o.      s  _  ^        .V    ,  H 
''^  ~  3  -  12  ~  15  "^  10 


^/i5.  a;  =  3-J,  y  ^  6f . 
Jw5.  a;  =  12,  y  =  6, 

Ans.  a;  =  2,  ^  =  —  1. 


22. 


42;  —  3.y  —  7  _  3a;      2y      5 
5  ~10       15       6 

?/— 1       a;      Sy_y—x      a;       11 
""3"  "^  2""20~"15~  "^6"^  10 


Ans.  x  =  3f  y=2. 


23.  ^ 


2a; 

6y 

3a; 

y 

3 

12 

2 

3 

7 

23 

4 

2 

X- 

1^ 

_1 

La;  +  2/      5 


Ans.  a;  =  18,  y  =  12. 


108 


SIMPLE    EQUATIONS. 


(12a;-    Gv  =  a    ) 


25. 


X 

+ 

t- 

2 

m 

71 

X 

U- 

1 

{m 

n 

Ans.  X  =  y  =:-. 


.  3m  n 

Ans.  a; r=  -;^ ,  y  =  ^. 


2  ' 


26. 


a       b 


3a       6b       3 


( 7nx  —  ny  =  a  ) 


Ans.  X  =:3a,  y  =  2b. 


en  -\-bd       _  cm  —  ad 
bm  -\-  an'  ^  ~  bra  +  an ' 


Ans.  X  =  r, : ,  y 


28.   - 


29.   - 


b-\-c 

a-\-c 

ax  — 

%_1 

{a- 

b)^c~' 

X 

a-\-b 

^  a-b 

x-y 
Aab 

=  1 

=  %a 


Ans.  x-=b  -\-  c,  y  =ia  -{•  c. 


Ans.  x={a  +  by,  y  =  {a  —  by. 


SIMPLE   EQUATIONS    WITH   AXT  NUMBER  OF  UNKNOWN 
QUANTITIES. 

212.  To  solve  a  group  of  simple  equations  contain- 
ing any  number  of  unknown  quantities. 

Let  it  be  required  to  solve  the  equations 

Sx-\-4:y-2z  =  10    .    .    .  (1), 

6a;  — 2?/ +  32  =  16    .     .    .  (2), 

4a;  +  2^  +  2;z  =  22    .     .    .  (3). 


AN^T    NUMBER    OF    UNKITOWIS"    QUANTITIES.  109  < 

Combining  (1)  and  (3),  also  (1)  and  (3),  eliminating  z  in  each 
case,  we  have  the  new  group 

19a; +  8?/ =  62    .     .    .     (4),  ^ 

7r?:  +  6?/  =  32    .     .     .     (5). 

Combining  (4)  and  (5),  eliminating  y,  we  have 
29ic  =  58; 
whence,  .  a;  =  2. 

Substituting  2  for  x  in  (5),  we  have 
14  +  6y  =  32; 
whence,  2/  =  ^• 

Substituting  2  for  x  and  3  for  y  in  (1),  we  have 
6  +  12  — 2;z  =  10; 
whence,  z  —  L 

B  ULE. 

I.  Cofnhine  one  equation  of  the  group  with  each  of  the  others, 
eliminating  the  same  unknoiun  quantity  in  each  case  ;  there  tuill 
result  a  new  group  containing  one  equation  less  than  the  original 
group. 

II.  Combine  one  equation  of  the  resulting  group  tvith  each  of 
the  others,  eliminating  a  second  unknown  quantity;  there  ivill 
result  a  new  group  containing  two  equations  less  than  the  original 
group. 

III.  Continue  the  operation  until  a  single  equation  is  found, 
contaiiiing  only  one  unknoiun  quantity. 

IV.  Find  the  value  of  this  unknown  quantity  hy  the  rule  of 
Art.  197;  substitute  this  value  in  either  one  of  the  group  of  tioo 
equations,  and  find  the  value  of  a  second  unknoimi  quantity; 
then  substitute  the  two  values  thus  fou7ul  in  any  one  of  the  group 
of  three  equations,  and  find  the  value  of  a  third  unknown  quan- 
tity;  and  so  on,  till  the  values  of  all  are  found. 


•110  SIMPLE    EQUATIONS. 

ScH.— When  any  one  of  the  unknown  quantities  does  not 
occur  in  all  the  equations,  it  will  generally  be  best  to  eliminate 
that  quantity  first. 

EXAMPLES. 

Solve  the  following  groups  of  simultaneous  equations : 

1.  )2x  —  dtj+z  =  iy'  Ans.  x  =  S,y  =  2yZ  =  l. 
(dx—   y  +  22  =  9  ) 

(dx-{'2t/-'4:Z  =  15) 

2.  }6x  —  3y-\-2z  =  2S>'  Ans.  x=il,y  =  b,z  =  L 

[dy  ^^—    x  =  2^) 

I    a;+    y—   z  =  l  ) 

3.  <8x  +  3y  —  6z  =  l>'  Ans.  x  =  2,  y  =  3^  z=z  4, 
i3z  —  4^—   y=l) 

(2x-7y  +  ^=   0) 

4.  l3x  —  3y-\--z=o['  Ans.  x=l,  y  =  2,  z  =  3. 
idx  +  by  -{-3z  =  28  ) 

iLx  —  3y-\-2z=9\ 

5.  <  2x  +  5y  —  3z  =    4  ?■  •  A^is.  x  =  2^  y  =z  3,  z  =  5. 
(  5a:  +  6?/  —  22  =  18  ) 

(2x-^   4:y  +    92  =  28  I 

6.  •]  7a;  +    3y  —   6z  =    3>,  Ans.  x  =  2,  y  =  3,  z  =  4:. 
i  2x  4-  lOy  —  ll2  =    4  ) 

(    x  —  2y-\-3z=    6  ^ 

7.  \2x-{-3y  —  4zz=i20y'    -  Ans.  x  =  S,y=z4,z  =  2. 
{3x  —  2y  +  5z  =  2Q) 

(4^-^3y  +  2z  =  4L0) 

8.  -j  5a;  +  9?/  —  72  =  47  [■  •  Ans.  x  =  10,y  =  2j  2=3. 
(  9a;  +  82/  —  32  =  97  ) 

(    Sx-\-2y  -{-    z  =  23) 

9.  I    5a;  +  2z/  +  42  =  46  y  .  A71S.  x  =  4=,  y  =  3,  z  =  0, 
(lOa;  +  5?/  +  42  =  75  ) 


ANY    NUMBER    OF    UNKNOWN    QUANTITIES. 


Ill 


10. 


11. 


12. 


13. 


14. 


16. 


17. 


DX  —  6y  -\-  4:Z  =z  15 
7a;  -\.4:y  —  dz  =  ld 
^^  +  y  +  Qz  =  ^6 

x-{-y  -{-z  =  31)' 
X  -j-  y  —  z  =  26  [' 
X  —  y  —  z=    9; 

x-}-y^z  =  26) 
X  —  y  =     4  V  • 

X—  z         =    6 ) 

x  —  y  —  z=    (j\ 
3y  —  x  —  z  =  12\' 
'^z  —  y  —  x  =  24:) 


Ans.  X  =z  S,  y  =  4:,  z  =z  6, 
Ans,  ic  =  20,  y  =  S,  z^3. 
Ans.  X  =  12j  y  =  S,  z  =  6. 
Ans.  X  =  39, «/  =  21,  z  =  12. 


t3y-l 
4       ~ 

6z 
5 

X 

2 

-1 

5x      ^ 

=  y  + 

5 

6 

3a: +  1 

L        7 

z 
14  + 

1 
6 

=1- 

"  10a;  4-  4y 

-6z 

= 

4a;  +  Gy 

5 

9 

Ans,  x=2,  y=3j  z=l. 


3z 


10a;  -]-  4:y  —  5z  =  4x  -h  6y  —  3z  —  8 

10a;  -{-  4ry  —  5z      4x  -\-  Qy  —  3z x  -^  y  ■{■  z 


10 


+ 


3 


20 


46 


Ans.  x  =  e,y  =  —,z  =  Y' 


'7x-3y  =  l} 
llz  —7u  =  l 

4z  —7y  =  1 
19a;  —  3w  =  1 

"3u  —  2y=    2 

5x  —  7z  =  11 

2a;  +  3y  =  39 

I4:y  -\-3z  =4:1  ) 


Ans.  x  =  4,  y  =  0,  z  =  IGf  u  =  'Z5« 


Ans.  X  =  12,  y  =  5,  z  z=  7,  u  zzz  4. 


112 


SIMPLE    EQUATIONS. 


18. 


19. 


2x 


37/  -\-2z  =  13 
4:1/  -\-2z  =  14 
4:U  —  2x  =  30 
5y  +  ^u  =  32  J 


Ans.  x=3,  ?/=!,  z=5,  u=zd. 


Hu  —  13z  =  87 

lOy—    3a;  =  11 

Sic  +  14a;  =  57 

2x  —  llz  =  50 


>■ .  Ans.  x  =  d,  y  =  2,  z=:  —  4:,  uz=6. 


['^x-2z  +  Su  =  17  ^ 
4:y  —  2z  +    ?;  =  11 

20.  ^  5y  —  3x  —  2u=z  8 
%  —  3?^  4-  2v  =  9 
3z  -^Su  =33 

^W5.  a;  =  2,  ?/  =  4,  2  =  3,  ?^  =  3,  ?;  =  1. 

'  3a;  —  4?/  +  32J  +  3t;  —  Gu  =  11  ^ 
3a;  —  5?/  4-  2z  —  4w  =  11 

21.  ^  lOy  —  3z+3u  —  2v=  2 
5;z  +  4w  +  2v  —  2a;  =  3 
Qu  —  3v  -{-  4:X  —  2y  =    6 

^?i5.  a;  =  2, 2/  =  1, 2;  =  3, 2^  =  —  1,  ?;  =  —  2. 


X      y      ^ 
22.  ^  ?  +  i  =  1 

0       c 


(ay+I}x  =  c) 
23.     -^  ca;  +  a^;  =  5  )■ . 
\  iz  -\-  cy  =:  a) 

Ans.  X  = :tt ,  y 

2bc  ^ 


.  a  h  c 

Ans.  x=z^,  y--^,  ZZZZ-, 


2ac 


z  = 


a^  +  l?  —  (f 
2ab 


r  a;  +  «=   ?/+    z 

24.     K  2/  4-  fl^  =  2a;  +  25? 

(  z  +  a  =  3x  -]-3y 


Ans.  X 


-a,y^-^a,z  =  -a. 


ANY    NUMBER    OF    UNKNOWN    QUANTITIES. 


113 


25.     I  {b-^c)x+{a  +  c)y+  (a  +  b)z 
i  hex  +  acy  +  abz  =  1 

1  1 


Ans.  X  = 


V  y 


26. 


27. 


28. 


29. 


30. 


(/f-a){b-cy 


(a—b){a—c) 

ax  -\- hy  -\-  cz  =:  A.  \ 

a^x  4-  l^y  +  c^^  =  A2  V  .       Ans.  x 

a^x  -\-  l^y  +  c^^  =  A^ ) 

x-\-y-\-z  =  a  +  h-\-c  \ 

hx  -\-  cy  -\-  az  ^cx  -{-  ay  +  hz\  . 
ex  -\-  ay  -\-  iz  =  a^  -}-  b^  +  c^     ) 

X  —  ay  -^  a^z^a^  \ 
x—by+b^z  =  b^>. 
X  —  cy  -]-  ch  =  c^  ) 

ex  -\-  y  -}-  az  =  2a  ) 

c^x  -\-  y  -}-  ah  =  2ac         Y  . 
aex  —  y  +  aez=za^-{-c^) 

Ans.  x=z 


(c—a){c—b) 


A{A-b){A~  c) 
a(a  —  b)  {a  —  c) 


Ans.  x  =  b  +  e 


Ans.  X  =  abc. 


a-\-l 


y=ia  —  e,  2;  = 


^  It  +  V  -\-  to  -\-  X  -\-  y  ^16 
V  -\-w  +  x-^y-{-z  =20 
w  +x  -{-  y  -{■  z  -\-  u  =19 
X  -\-  y  -\-z  -^u  -{-V  =1^ 
y  -\-  z  -{-ti  -\-v  -\-w  =  l'^ 

^Z    -\-  U    -\-  V   -{•  W  -\-  X  =z\^  ) 

Ans.  u  =  l,  1;  =  2,  ?(;  =  3,  ic  =  4,  y  =  6,  z=6. 


213. 


FROBIjEMS. 


1.  A  and  B  engage  in  play ;  in  the  first  game  A  wins  as  much 
as  he  had  and  four  dollars  more,  and  finds  he  has  twice  as  much 
as  B ;  in  the  second  game  B  wins  half  as  much  as  he  had  at  first 
and  one  dollar  more,  and  then  it  appears  he  has  three  times  as 
much  as  A ;  what  sum  had  each  at  first  ? 

Let  X  =  the  number  of  dollars  which  A  had,  and 
y  =  the  number  of  dollars  which  B  had; 


114  SIMPLE    EQUATIONS. 

then  after  the  first  game  A  has  2a;  4-4  dollars,  and  B  has  y—x—^ 
dollars. 

.  • .     by  the  first  condition, 

2a;  -I-  4  =  2  (?^  -  a;  -4)    .    .     .     (1). 
Again,  after  the  second  game  A  has  2a;  +  4  —  *^  —  1  dollars, 
and  Bhas?/  —  a;  —  4  +  ^-fl  dollars. 
.  • .    by  the  second  condition, 

y-a;-4  +  |  +  l=3(2.T  +  4-|-l)     .    .    .(2). 
By  transposition  and  reduction,  (1)  and  (2)  become, 


y  —  2x=    6     .     .     . 

(3), 

3y  -  7a;  =  12    .    .    . 

(4). 

Multiplying  (3)  by  3, 

3?/  —  6a?  =  18     .    . 

.     (5). 

Subtracting  (4)  from  (5), 

a;  =  6. 

Substituting  6  for  x  in  (3),  we  find 

2/ =  18. 

2.  A  sum  of  money  was  divided  equally  among  a  certain  num- 
ber of  persons ;  had  there  been  three  more  persons,  each  would 
have  received  one  dollar  less,  and  had  the  number  of  persons  been 
two  less,  each  would  have  received  one  dollar  more  than  he  did ; 
what  was  the  number  of  persons,  and  what  did  each  receive  ? 

Let  X  =  the  number  of  persons,  and 

y  =  the  number  of  dollars  each  received; 
then  xy  dollars  is  the  sum  divided. 

By  the  conditions  of  the  problem,  the  sum  divided  is  also  ex- 
pressed by  (x  +  3)  (y  -  1),  or  (a;  -  2)  (2/  +  1) ;  .-.  (43,  6), 
we  have. 


AN^Y    NUMBER    OF    UKKKOWN    QUAi^TITIES.  115 

{x-\-2){y-l)=xy     .    .    .     (1), 

{x-^)(y  +  l)=xy    .    .    .     (2). 

By  transposition  and  reduction,  (1)  and  (2)  become 

Zy-x  =  Z     .    .     .     (3), 

x-^  =  2     .     .    .     (4). 

Eliminating  x  from  (3)  and  (4), 

^  —  '^y  —  5,  or 
«/  =  5; 
.-.  by  (4),  a;  rrz  2?/  +  2  =  10  +  2  =  12. 

3.  What  fraction  is  that  which  becomes  equal  to  J  when  its 
numerator  is  increased  by  6,  and  equal  to  \  when  its  denominator 
is  diminished  by  2  ? 

Let  a;  =  the  numerator,  and 

y  =  the  denominator  of  the  fraction ; 
then,  by  the  conditions  of  the  problem, 

3 


y    -^    '    '    ' 

W> 

X             1 

y-%-2      '     '     ' 

(2). 

Clearing  of  fractions,  transposing  and 

reducing, 

3y  —  4:X  =  24:        .       . 

•     (3), 

y-2x  =  2     .    . 

•     (4). 

Multiplying  (4)  by  2,  and  subtracting  the  result  from 

(3),  we 

y  =  20; 

.-.    by  (4),                       x=    9. 

Therefore  the  required  fraction  is  ^. 

4.  Find  two  numbers  whose  sum  is  «,  and  whose  difference 
is  b. 


116  SIMPLE    EQUATIONS. 

Let  a;  =  the  gi*eater  number,  and 
y  =  the  less  number ; 
then,  by  the  conditions  of  the  problem, 


x-\-y  =  a    .    .    . 
x  —  y  =  b    .    .    . 

(1), 

(3); 

whence, 

a  ,   b 

^  =  2  +  2' 

and 

a      b 

Since  a  and  b  are  any  numbers  whatever,  we  have  the  follow- 
ing general  principles,  by  means  of  which  all  similai*  problems  can 
be  solved : 

1.  The  greater  of  two  numbers  is  found  by  adding  half  their 
differe7ice  to  half  their  sum. 

2.  TJie  less  of  two  numbers  is  found  by  subtracting  half  their 
difference  from  half  their  sum. 

5.  A  and  B  together  possess  $570.  If  A's  money  were  three 
times  what  it  really  is,  and  B's  five  times  what  it  really  is,  the  sum 
would  be  ^2350.     How  much  money  does  each  possess  ? 

Ans.  A  $250,  B  $320. 

6.  Find  two  numbers  such  that  if  the  first  be  added  to  four 
times  the  second,  the  sum  is  29 ;  and  if  the  second  be  added  to 
six  times  the  first,  the  sum  is  36.  Ans.  5  and  6. 

7.  K  A's  money  were  increased  by  $36,  he  would  have  three 
times  as  much  as  B ;  but  if  B's  money  were  diminished  by  $5,  he 
would  have  half  as  much  as  A.     How  much  has  each  ? 

Ans.  A  f542,  B  |26. 

8.  A  and  B  lay  a  wager  of  $10 ;  if  A  loses,  he  will  have  $25 
less  than  twice  as  much  as  B  will  then  have;  but  if  B  loses,  he 
will  have  five-seventeenths  of  what  A  will  then  have.  How  much 
money  has  each?  Ans.  A  $75,  B  835. 

9.  For  $21,  either  32  pounds  of  tea  and  15  pounds  of  coffee, 
or  36  pounds  of  tea  and  9  pounds  of  coffee,  can  be  bought ;  find 
the  price  per  pound  of  each.       Ayis.  Tea  50  cts.,  coffee  o^\  cts. 


ANY    NUMBEE    OF    UNKNOWN    QUANTITIES.  117 

10.  A  pound  of  tea  and  three  pounds  of  sugar  cost  $1.20;  but 
11  lea  were  to  rise  50  per  cent,  and  sugar  10  per  cent.,  they  would 
cost  $1.56 ;  find  the  price  per  pound  of  each. 

Ans.  Tea  60  cents,  sugar  20  cents. 

11.  A  and  B  together  can  perform  a  piece  of  work  in  8  days, 
A  and  C  together  in  9  days,  and  B  and  0  in  10  days ;  in  what 
time  could  each  person  alone  perform  the  same  work  ? 

Ans.  A,  14}f  days;  B,  17f|;  C,  23^. 

12.  A  and  B  together  can  perform  a  piece  of  work  in  a  days, 

A  and  C  together  in  b  days,  and  B  and  C  together  in  c  days ;  in 

what  time  could  each  person  alone  perform  the  same  work  ? 

.  .  2abc  , 

A  in f 7  days, 

ac  +  oc  —  ao      *' 

Ans.    ^  B  m  -7 ^ days, 

ab  ■\-  be  —  ac     '' 

^  .  2abc  , 

C  m  —, r-  days. 

ab  -\-  ac  ^  be      '' 

13.  A  person  possesses  a  certain  capital,  which  is  invested  at  a 
certain  rate  per  cent.  A  second  person  has  $1000  more  than  the 
first,  and  investing  his  capital  one  per  cent,  more  advantageously, 
has  an  income  greater  by  $80.  A  third  person  has  $1500  more 
capital  than  the  first,  and  investing  it  two  per  cent,  more  advan- 
tageously, has  an  income  greater  by  $150.  Find  the  capital  of 
each  person  and  tlie  rate  at  which  it  is  invested. 

j  Sums  at  interest,  $3000,  $4000,  $4500. 

( Rates  of  interest,         4,         5,  6  percent. 

14.  If  there  were  no  accidents,  it  would  take  half  as  long  to 
travel  the  distance  from  A  to  B  by  railroad  as  by  coach ;  but 
three  hours  being  allowed  for  accidental  stoppages  by  the  former, 
the  coach  will  travel  all  the  distance  but  fifteen  miles  in  the  same 
time ;  if  the  distance  were  two-thirds  as  great  as  it  is,  and  the 
same  time  allowed  for  railway  stoppages,  the  coach  would  take 
exactly  the  same  time ;  find  the  distance  from  A  to  B. 

Ans.  90  miles. 

15.  A  and  B  are  set  to  a  piece  of  work  which  they  can  finish 
in  thirty  days,  working  together,  for  which  they  are  to  receive 


118  SIMPLE    EQUATIONS. 

$64.  When  the  work  is  half  finished,  A  rests  eight  days  and  B 
four  days,  in  consequence  of  which  the  work  occupies  five  and  a 
half  days  more  than  it  would  otherwise  have  done.  How  much 
ought  each  to  receive?  Aiis.  A  $22,  B  142. 

16.  A  and  B  run  a  mile.  First  A  gives  B  a  start  of  44  yards, 
and  beats  him  by  51  seconds;  at  the  second  heat  A  gives  B  a 
start  of  1  minute  and  15  seconds,  and  is  beaten  by  88  yards.  In 
what  time  can  each  run  a  mile  ? 

Ans.  A  in  5  minutes,  B  in  6  minutes. 

17.  A  and  B  start  together  from  the  foot  of  a  mountain  to 
go  to  tlie  summit.  A  would  reach  it  half  an  hour  before  B, 
but,  missing  his  way,  goes  a  mile  and  back  again  needlessly,  dur- 
ing which  he  walks  at  twice  his  former  pace,  and  reaches  the  top 
six  minutes  before  B.  C  starts  twenty  minutes  after  A  and  B, 
and,  walking  at  the  rate  of  two  and  one-seventh  miles  per  hour, 
arrives  at  the  summit  ten  minutes  after  B.  Find  A's  and  B's 
rates  of  walking,  and  the  distance  from  the  foot  to  the  summit  of 
the  mountain.         Ans.  2^,  2  miles  per  hour;  distance,  5  miles. 

18.  A  number  expressed  by  two  digits  is  four  times  the  sum  of 
the  digits,  and  if  27  be  added  to  the  number  the  order  of  the 
digits  will  be  inverted;  find  the  number. 

Let  X  =  the  left  digit,  and 
y  =  the  right  digit; 
then,  since  x  stainds  in  the  place  of  tens,  the  number  will  be  rep- 
resented by  10^  +  y. 

.*.    by  the  first  condition, 

10x  +  y  =  4.{x-{-y).    .    .    (1); 

and  by  the  second  condition, 

lOx  +  y  -{- 211  =  lOy  -}-x.    .    ,    (2). 

Solving  these  equations,  we  find 

x  =  3f    and    y  =  6; 

.  • .    lOo:  4-  y  =  30  4-  6  =  36,  the  number  required. 

19.  A  number  is  expressed  by  three  digits.  The  middle  digit 
is  equal  to  twice  the  left-hand  digit,  and  greater  by  3  than  the 


ANY    NUMBEK    OF    UKKNOWN    QUANTITIES.  119 

right-hand  digit.    If  99  be  subtracted  from  the  number,  the  order 
of  the  digits  will  be  inverted;  find  the  number.  Ans.  241. 

20.  A  number  consisting  of  two  digits  contains  the  sum  of  its 
digits  four  times,  and  their  product  three  times;  find  the  number. 

Ans.  24. 

21.  A  railway  train,  after  travehng  for  one  hour,  meets  with 
an  accident  which  delays  it  one  hour,  after  which  it  proceeds  at 
three-fifths  of  its  former  rate,  and  arrives  at  the  terminus  three 
hours  behind  time;  had  the  accident  occurred  50  miles  further 
on,  the  train  would  have  arrived  1  hour  and  20  minutes  sooner; 
find  the  length  of  the  line,  and  the  original  rate  of  the  train. 

Ans.  100  miles;  original  rate,  25  miles  per  hour. 

22.  A  railway  train,  running  from  London  to  Cambridge,  meets 

with  an  accident  which  causes  it  to  diminish  its  speed  to -th 

n 

of  what  it  was  before,  in  consequence  of  which  it  is  a.  hours  late. 

If  the  accident  had  occurred  b  miles  nearer  Cambridge,  the  train 

would  have  been  c  hours  late.     Find  the  original  rate  of  the  train. 

Ans.  — miles  per  hour. 

a  —  c  ^ 

23.  The  fore-wheel  of  a  carriage  makes  six  revolutions  more 
than  the  hind-wheel  in  going  120  yards.  If  the  circumference  of 
the  fore-wheel  be  increased  by  one-fourth  of  its  present  size,  and 
the  circumference  of  the  hind-wheel  by  one-fifth  of  its  present 
size,  the  six  will  be  changed  to  four.  Required  the  circumference 
of  each  wheel.  A71S.  4  yards  and  5  yards. 

24.  A  man  starts  p  hours  before  a  coach,  and  both  travel  uni- 
formly ;  the  latter  passes  the  former  after  a  certain  number  of 
hours.  From  this  point  the  coach  increases  its  speed  to  six-fifths 
of  its  former  rate,  while  the  man  increases  his  to  five-fourths  of 
his  former  rate,  and  they  continue  at  these  increased  rates  for 
g  hours  longer  than  it  took  the  coach  to  overtake  the  man.  They 
are  then  92  miles  apart;  but  had  they  continued  for  the  same 
length  of  time  at  their  original  rates,  they  would  have  been  only 
80  miles  apart.  Show^  that  the  original  rate  of  the  coach  is  twice 
that  of  the  man.  Also,  if  p  +  q  =  1G,  show  that  the  original 
rate  of  the  coach  was  10  miles  per  hour. 


120 


SIMPLE    EQUATIONS. 


314. 


SYNOPSIS    FOR    REVIEW. 


CO   Eh 


Members  . 


Kinds 


First  Member. 
Second  Member. 

Identical  Equation. 
Equation  of  Condition. 
Numerical  Equation. 
Literal  Equation. 


Klnds  respecting  \    *^^,  '  . 
Degree.  U^^adratw. 


'  Higher 


C  Cubic. 

\  Biquadratic,  etc. 


Transformations. 


{  Clearing  of  Fractions.    Rule. 


(  Transposition  of  Terms.     Rule.    Cor. 

Solution  of   Simple  Equations  containing  only  one  Un- 
known Quantity. 


Solution  of  Pros. 


Simultaneous 
Equations. 


Statement. 

Solution  of  Equation. 

Rule. 

fEliminav'on    by  addi" 
tion  or  subtraction. 
Rule.   Sch.  1,  2,3.4. 
EliminatioD  by  substi- 
tution.    Rule. 
Elimination    by  com- 
parison.    Rule. 
, General  Sch. 

Rule  for  groups  of  Equations  containing  any 
number  of  unknown  quantities. 


'  Solution  for  two  un- 
known quantities. 


DISCUSSION  OF  PROBLEMS  LEADING  TO  SIMPLE  EQUATIONS. 


215.  After  a  problem  has  been  solved,  we  may  inquire  what 
values  the  unknown  quantities  will  have,  when  particular  suppo- 
sitions are  made  with  regard  to  the  given  quantities.  The  deter- 
mination of  these  values,  and  their  interpretation,  constitute  the 
Discussion  of  the  I*rohlem, 


DISCUSSIOK    OF    PROBLEMS.  121 

216.       IKTEEPRETATION"   OF  K^EGATIVE   RESULTS. 

1.  What  number  must  be  added  to  a  number  a  in  order  that 
the  sum  may  be  ^  ? 

Let  X  =  the  required  number ;  then,  by  the  question, 

a  +  ic  =  $; 

whence.  x=ih  —  a. 

This  is  a  general  solution,  a  and  b  being  arbitrary  quantities. 
If  a=^  12,  and  b  =  25,  we  have 

a;  =  25  —  12  =  13. 

But  suppose  a  =  30,  and  b  =  24=;  then 
a;  =  24  —  30  =  —  6. 

How  is  this  negative  result  to  be  interpreted  ? 

If  we  recur  to  the  enunciation  of  the  problem,  we  see  that  it 
now  reads  thus :  What  number  must  be  added  to  30  in  order  that 
the  sum  may  be  24  ? 

Here  it  is  obvious  that  if  the  words  added  and  sum  are  to  re- 
tain their  arithmetical  meanings,  the  proposed  problem  is  impos- 
sible. But  we  see  at  the  same  time  that  the  following  problem 
can  be  solved :  What  number  must  be  taken  from  30  in  order  that 
the  difference  may  be  24  ?     The  answer  to  this  problem  is  6. 

The  second  enunciation  differs  from  the  first  in  these  respects : 
The  words  added  to  are  replaced  by  taken  from,  and  the  word 
sicm  by  difference. 

Hence  we  may  say  that,  in  this  example,  the  negative  result 
indicates  that  the  problem,  in  a  strictly  arithmetical  sense,  is  im- 
possible ;  but  that  a  new  problem  can  be  formed  by  appropriate 
changes  in  the  original  enunciation,  to  which  the  absolute  value 
of  the  negative  result  will  be  the  correct  answer. 

This  indicates  the  convenience  of  using  the  word  add,  in  Alge- 
bra, in  a  more  extensive  sense  than  it  has  in  Arithmetic. 

Let  X  denote  a  quantity  which  is  to  be  added  algebraically  to 
a ;  then  the  algebraic  sum  is  a  -{-  x,  whether  x  be  positive  or  neg- 
ative. 

Hence,  the  equation  a  -\-  x  =  b  will  be  possible  algebraically, 
whether  a  be  greater  or  less  than  b. 


122  SIMPLE    EQUATIONS. 

2.  A's  age  is  a  years,  and  B's  age  is  l  years;  when  will  A  be 
twice  as  old  as  B? 

Let  X  =  the  required  number  of  years ;  then,  by  the  question, 

a-\-x=:2{b  -\-x); 

whence,  x=a  —  2b. 

K  a  =  40  and  b  =  15,  then 

re  =  40  —  30  =  10. 

But  suppose  a  =  35  and  h  =  20,  then 

a;  =  35  —  40  =  —  5. 

Here,  as  in  the  preceding  problem,  we  are  led  to  inquire  into 
the  meaning  of  the  negative  result.  Now,  with  the  assigned 
values  of  a  and  Z>,  the  equation  which  we  have  to  solve  becomes 

35  4-  a;  =  40  +  2x. 

This  equation  is  impossible,  if  a  strictly  arithmetical  meaning 
is  to  be  given  to  the  symbols  x  and  + ,  for  40  is  greater  than  35, 
and  2x  is  greater  than  x.  But  let  us  change  the  enunciation  to 
the  following :  A's  age  is  35  years,  and  B's  age  is  20  years ;  when 
tvas  A  twice  as  old  as  B  ? 

Let  x  =  the  required  number  of  years ;  then,  by  the  question, 

35  —  re  =  2  (20  —  a:)  =  40  —  2rr; 
whence,  x  =  5. 

Here  again,  we  may  say  the  negative  result  indicates  that  the 
problem,  in  a  strictly  arithmetical  sense,  is  impossible  ;  but  that  a 
new  problem  can  be  formed  by  appropriate  changes  in  the  original 
enunciation,  to  which  the  absolute  value  of  the  negative  result 
will  be  the  correct  answer. 

We  may  observe  that  the  equation  corresponding  to  the  new 
enunciation  may  be  obtained  from  the  original  equation  by 
changing  x  into  —  x. 

Suppose  the  problem  had  been  originally  enunciated  thus: 
A's  age  is  a  years,  and  B's  age  is  b  years;  find  the  epoch  at  which 
A's  age  is  twice  that  of  B. 


DISCUSSION    or    PROBLEMS.  123 

We  cannot  tell  from  the  enunciation  of  the  problem  whether 
tlie  required  epoch  is  before  or  after  the  present  date.  If  we  sup- 
pose the  required  epoch  to  be  x  years  after  the  present  date,  we 

obtain 

x-=a  —  %l). 

If  we  suppose  the  required  epoch  to  be  x  years  lefore  the 
present  date,  we  obtain 

x:=^%b—a. 

If  25  is  less  than  a,  the  first  supposition  is  correct,  since  it 
leads  to  a  positive  value  for  x ;  the  second  supposition  is  incorrect, 
since  it  leads  to  a  negative  value  for  a*. 

If  V)  is  greater  than  a,  the  second  supposition  is  correct,  since 
it  leads  to  a  positive  value  for  x\  the  first  supposition  is  incorrect, 
since  it  leads  to  a  negative  value  for  x. 

Here  we  may  say,  then,  that  a  negative  result  indicates  that 
we  made  the  wrong  choice  out  of  two  possible  suppositions  which 
the  problem  allowed.  But  it  is  important  to  notice,  that  when 
we  discover  that  we  have  made  the  wrong  choice,  it  is  not  neces- 
sary to  go  through  the  whole  investigation  again,  for  we  can  make 
use  of  the  result  obtained  on  the  wrong  supposition.  We  have 
only  to  take  the  absolute  value  of  the  negative  result  and  place 
the  epoch  before  the  present  date  if  we  had  supposed  it  after,  or 
after  the  present  date  if  we  had  supposed  it  before. 

3.  A's  age  is  a  years,  and  B's  age  is  h  years ;  when  was  A  twice 
as  old  as  B  ? 

Let  X  =  the  required  number  of  years ;  then 

a  —  x:=2{b  —  x)', 
whence,  x  =  2b  —  a. 

Now  let  us  verify  the  solution.    Substituting  2J  —  «  for  x, 

we  have 

a  —  X  =  a  —  {2b  —  a)  =  2a  —  2b; 

and  2{b  —  x)  =  2{b-'2b-^a)  =  2a  —  2b. 

If  b  is  less  than  «,  these  results  are  positive,  and  there  is  no 
arithmetical  difiiculty.    But  if  b  is  greater  than  «,  although  the 


124  SIMPLE    EQUATIONS. 

two  members  are  algebraically  equal,  yet,  since  they  are  both  neg- 
ative quantities,  we  cannot  say  that  we  have  arithmetically  veri- 
fied the  solution ;  and  when  we  recur  to  the  problem,  we  see  that 
it  is  impossible  if  a  is  less  than  i ;  because,  if  at  a  given  date  A's 
age  is  less  than  B's,  then  A's  age  never  was  twice  B's,  and  never 
will  be. 

Or,  without  proceeding  to  verify  the  result,  we  may  observe 
that  if  d  is  greater  than  a,  then  x  is  also  greater  than  a,  which  is 
inadmissible. 

Thus  it  appears  that  a  problem  may  be  really  absurd,  and  yet 
the  result  may  not  immediately  present  any  difficulty,  though 
when  we  proceed  to  examine  or  verify  this  result,  we  may  discover 
an  intimation  of  the  absurdity. 

The  equation 

a  4-  a:  =  2  (J  +  a:) 

may  be  considered  as  the  symbolical  expression  of  the  following 
verbal  enunciation : 

Suppose  a  and  J  to  be  two  quantities ;  what  quantity  must  be 
added  to  each,  so  that  the  first  sum  may  be  twice  the  second  ? 

Here  the  words  quantity,  sum,  and  added  may  all  be  under- 
stood in  algebraic  senses,  so  that  x,  a,  and  J  may  be  positive  or 
negative. 

This  algebraic  statement  includes  the  arithmetical  question 
about  the  ages  of  A  and  B. 

It  appears,  then,  that  when  we  translate  a  problem  into  an 
equation,  the  same  equation  may  be  the  symbolical  expression  of 
a  more  comprehensive  problem  than  that  from  which  it  was  de- 
rived. 

When  the  solution  of  a  problem  leads  to  a  negative  result, 
and  the  student  wishes  to  form  an  analogous  problem  that 
shall  lead  to  the  corresponding  positive  result,  he  may  proceed 
thus: 

Change  x  into  —  x  m  the  equation  that  has  been  obtained, 
and  then,  if  possible,  modify  the  verbal  statement  of  the  problem, 
so  as  to  make  it  coincident  with  the  new  equation. 

We  say  if  j^ossible,  because  in  some  cases  no  such  verbal  mod- 
ification seems  attainable,  and  the  problem  may  then  be  regarded 


(  DISCUSSION    OF    PROBLEMS.  125 

as  altogether  impossible.    To  illustrate,  take  the  following  prob- 
lem: 

4.  A's  age  is  20  years,  and  B's  age  is  30  years ;  when  will  the 
age  of  A  be  twice  that  of  B  ? 

Let  X  =  the  required  number  of  years ;  then 

20  +  2;  =  2  (30  +  a;)  =  60  +  2a;; 

whence,  a;  =  —  40. 

This  negative  result  shows  that  the  epoch  is  not  in  the 
future.  Suppose  it  to  be  in  the  past.  Changing  x  into  —  x,  the 
original  equation  becomes 

20  —  a:  =  2  (30  —  a;) ; 

whence,  x  =  40. 

This  result  seems  to  indicate  that  40  years  ago — that  is,  20 
years  before  A  was  bom,  and  10  years  before  B  was  born—  A  was 
twice  as  old  as  B.  A  manifest  absurdity.  Hence,  the  problem  is 
an  impossible  one. 

Prii^ciples — 1.  A  negative  result  may  arise  from  the  fact 
that  the  problem,  contains  a  condition  ivhich  cannot  he  arithmet- 
ically satisfied  ;  or  from  the  fact  that,  of  tivo  possible  suppositions 
respecting  the  quality  of  a  quantity,  ive  adopted  the  wrong  one. 

2.  After  a  problem  has  been  translated  into  an  equation,  the 
qualify  of  any  quantity  involved  will  be  changed,  if  we  change  the 
sign  of  the  symbol  of  that  quantity, 

PJROBLJE31S. 

1.  A  father's  age  is  40  years,  and  his  son's  age  is  13  years ; 
when  will  the  age  of  the  father  be  four  times  that  of  the  son  ? 

Ans.  x  =  —  4:. 
Modify  the  enunciation  so  that  the  result  shall  be  +  4. 

2.  Find  two  numbers  whose  sum  is  2  and  difference  8. 

Ans.  —  3  and   -|-  5. 
Modify  the  enunciation  so  that  the  result  shall  be  +3  and  +5. 


126  SIMPLE    EQUATIONS. 

3.  The  difference  of  two  numbers  is  6,  and  four  times  the 
exceeds  five  times  the  greater  hy  12 ;  find  the  numbers. 

Ans.  —  42  and  —  36. 
Modify  the  enunciation  so  that  the  result  shall  be   -\-  42  and 
+  36. 

4.  Two  men,  A  and  B,  began  trade  at  the  same  time,  A  having 
three  times  as  much  money  as  B.  When  A  had  gained  $400  and 
B  1150,  A  had  twice  as  much  money  as  B ;  how  much  did  each 
have  at  first  ?  Ans,  A  was  in  debt  $300,  and  B  $100. 

Modify  as  in  the  preceding  examples. 

5.  There  are  two  numbers  whose  difference  is  a ;  and  if  three 
times  the  greater  be  added  to  five  times  the  less,  the  sum  will  be 
b.    What  are  the  numbers  ? 

.        b  -\-  5a         ,     b  —  Sa 
Ans.  — - —    and    — - — . 

o  o 

Interpret  this  result  when  a  =  24  and  b  =  48. 

6.  Two  men  were  traveling  on  the  same  road  toward  Boston, 
A  at  the  rate  of  a  miles  per  hour,  and  B  at  the  rate  of  b  miles  per 
hour.  At  6  o'clock  a.m.  A  was  at  a  point  m  miles  from  Boston, 
and  at  10  o'clock  a.m.  B  was  at  a  point  n  miles  from  Boston. 
When  did  A  pass  B  ? 

A71S.  7 —  hours  after  6  o'clock  a.m. 

a  —  b 

If  m  =  36,  n  =  28,  a  =  5,  and  5  =  3,  at  what  time  did  A 

pass  B? 

ZERO   AND   INFINITY.     FINITE,  DETERMINATE,  AND  INDETER- 
MINATE QUANTITIES. 

217.  The  symbol  0,  called  Nothing,  or  Zero,  is  used  to 
denote  the  absence  of  value,  or  to  represent  a  quantity  less  than 
any  assignable  value. 

A  quantity  less  than  any  assignable  value  is  sometimes  called 
an  Inftnitesinial, 

218.  The  symbol  00 ,  called  Infinity^  is  used  to  represent  a 
quantity  greater  than  any  assignable  value. 


ZERO    AND    INFIlflTY.  127 

219.  A  Finite  Quantity  is  one  whose  absolute  value  is 
comprisea  between  the  limits  0  and  oo . 

220.  A  Determinate  Quantity  is  one  which  has  only 
SL  finite  number  of  values. 

221.  An  Indeterminate  Quantity  is  one  which  has 
an  infinite  number  of  values. 


A  A  0  0 

INTERPRETATION    OF    THE    FORMS    — ,  — ,  -r-,  -,    oo  X  0, 

U  00  j^  y) 

AND   00  —  00  . 


00 


222.  In  order  to  explain  the  meaning  of  these  symbols,  let 
us  consider  the  fraction  :jr-. 

1.  Suppose  A  to  be  finite,  and  to  remam  unchanged,  while  B 
continually  decreases ;  then  the  value  of  the  fraction  -^  will  con- 
tinually  increase. 

Thus:  If  B  =  A;     then  ^=lj 
IfBz=:|;     then  ^  =  2; 
IfB  =  A;    then  1  =  10; 
If  B  =  A;  then  ^  =  100; 


Hence,  it  is  evident  that,  when  B  becomes  less  than  any  assign- 
e  quantity,  the  frac 
able  quantity ;  hence, 


able  quantity,  the  fraction  ^  will  become  greater  than  any  assign- 


A 


2.  If  the  denominator  B  is  made  to  increase  continually,  while 
the  nunjerator  A  remains  unchanged,  then  the  value  of  the  frac- 


128  SIMPLE    EQUATIONS. 

tion  rrj  will  continually  decrease ;  and  when  the  denominator  B 

becomes  greater  than  any  assignable  quantity,  the  fraction  will 
become  less  than  any  assignable  quantity ;  hence, 

00 

3.  If  the  numerator  A  is  made  to  decrease  continually,  while 
the  denominator  B  remains  unchanged,  then  the  value  of  the 

fraction  .^  will  continually  decrease ;  and  when  A  becomes  less 

than  any  assignable  quantity,  the  fraction  will  also  become  less 
than  any  assignable  quantity ;  hence, 

1  =  0. 

4.  Multiplying  both  members  of  the  equation  ^  =  0  by  B,  we 

have 

0  =  B  X  0. 

Dividing  both  members  of  this  equation  by  0,  we  have 

But  B  is  any  finite  quantity ;  hence  tt  is  a  Symbol  of  In~ 

det elimination  (221). 

6.  Multiplying  both  members  of  the  equation  —  =  0  by  c» ,  we 

have 

A  =  oo  X  0. 

But  A  is  any  finite  quantity ;  hence  oo  x  0  is  a  symbol  of  in- 
determination. 

6.  We  may  place  the  equation  x  =  B  under  the  following 

form: 

1 

0 


INFINITESIMALS    AND    INPINITIES.  129 


But  7:  =  ^  ;  hence, 


00  -D  . 


therefore  ^o  is  a  symbol  of  indetermination. 

7.  In  the  identity ^  =  — r-   make  o^  =  0  and  5  =  0 ;  we 

-^    a       b         ab  ^ 

then  have 

1_1_0 

0      0"~  0 

that  is,  00  —  00  =  - ; 

hence  oo  —  oo  is  a  symbol  of  indetermination. 


ORDERS  OF  INFINITESIMALS   AND   INFINITIES. 

223.  An  Inflnitesinial  of  the  First  Order  is  one 

that  is  infinitely  small  in  comparison  with  a  finite  quantity; 
that  is,  so  small  that  it  may  be  contained  in  a  finite  quantity 
an  infinite  number  of  times.  An  Infiiiitesifnal  of  the 
Second  Order  is  one  that  is  infinitely  small  in  comparison 
with  an  infinitesimal  of  the  first  order.  An  Jnfinitesi' 
mat  of  the  Third  Ordet*  is  one  that  is  infinitely  small 
in  comparison  with  an  infinitesimal  of  the  second  order;  and 
so  on. 

In  order  to  illustrate,  let  us  consider  the  continued  identity 

1  _  X  _x^  _a^ 

Let  X  be  an  infinitesimal  of  the  first  order ;  then  -  =  oo ;  that 

'  X 

is,  1  is  infinitely  great   in   comparison  with   x.      Again,   since 

1        r?/  1  X 

-  =:  —,  and  -  =  00 ,  it  follows  that  -5  =  oo ;  that  is,  x  is  infinitely 

X  Xt  X  X 

great  in  comparison  with  x^\  but  x  is,  by  hypothesis,  an  infinitesi- 
mal of  the  first  order;  therefore  x^  is  an  infinitesimal  of  the  sec- 
ond order.  In  like  manner,  it  may  be  shown  that  7?  is  an  infi- 
nitesimal of  the  third  order,  and  so  on. 


130  SIMPLE    EQUATIONS. 

234.  Infinities  are  of  different  orders  also.  Let  x  be  an  infi- 
nitesimal of  the  first  order,  and  A  any  finite  quantity;  then, 

AAA 

—  =  00   ...  (1),  -2  =  00   ...  (2),  ;;5  =  00   •  •  •  (3),  and  so  on. 

Now  the  denominator  in  the  first  member  of  (1)  is  infinitely 
great  in  comparison  with  the  denominator  in  the  first  member  of 
(2) ;  therefore  the  second  member  of  (2)  is  infinitely  great  in  com- 
pai'ison  with  the  second  member  of  (1). 

In  hke  manner  it  may  be  shown  that  the  oo  in  (3)  is  infinitely 
great  in  comparison  with  the  oo  in  (2) ;  and  so  on. 


235.  PROBLEM    OF    THE    COURIERS. 

The  discussion  of  the  following  problem,  originally  proposed 
by  Clairaut,  will  serve  to  illustrate  some  of  the  preceding  prin- 
ciples : 

Two  couriers,  A  and  B,  were  traveling  along  the  same  road 
and  in  the  same  direction,  namely,  from  C  toward  C ;  the  former 
going  at  the  rate  of  m  miles  per  hour  and  the  latter  at  the  rate  of 
n  miles  per  hour.  At  12  o'clock,  A  was  at  P,  and  B  was  d  miles 
in  advance  of  A.     When  were  the  couriers  together  ? 

q: IP 12 !£ 2 

We  cannot  teU  from  the  enunciation  whether  the  couriers 
were  together  before  or  after  12  o'clock ;  but  in  order  to  effect  a 
statement  of  the  problem,  we  will  suppose  the  required  time  to  be 
after  12  o'clock.  We  must  then  regard  time  after  12  o'clock  as 
positive,  and  time  before  12  o'clock  as  negative. 

Suppose  R  to  be  the  point  where  the  couriers  met,  and  Q  to 
be  the  point  where  B  was  at  12  o'clock. 

Let  X  =  the  required  number  of  hours ;  then,  since  A  traveled 

at  the  rate  of  m  miles  per  hour,  and  B  at  the  rate  of  n  miles  per 

hour,  we  have 

PR  =  mx^    and    QR  =  nx. 

But  PR  =  PQ  +  QR; 

mx  =z  d  -\-  7ix', 


whence,  x  = 


DISCUSSION.  131 

d 


m  —  n 


DISCUSSION. 
I.    Suppose  771  >  71. 

Under  this  hypothesis  the  vakie  of  x  will  be  positive,  because 
the  denominator  m  —  7i  is  positive.  Now,  since  x  is  positive,  we 
infer  that  the  couriers  were  together  after  12  o'clock. 

This  conclusion  is  consistent  with  the  conditions  of  the  prob- 
lem. For,  the  supposition  is  that  A  was  traveling  faster  than  B. 
A  would  therefore  gain  upon  B,  and  overtake  him  some  time  after 
12  o'clock. 

n.  Suppose  m  <  w. 

Under  this  hypothesis  the  value  of  x  will  be  negative,  because 
the  denominator  m  —  n  is  negative.  This  imphes  that  the  cou- 
riers were  together  before  12  o'clock. 

This  interpretation,  also,  agrees  with  the  conditions  of  the 
problem  under  the  present  hypothesis.  For,  since  m  <,7i,B  was 
traveling  faster  than  A ;  and,  as  B  was  in  advance  of  A  at  12 
o'clock,  he  must  have  passed  A  before  that  time. 

III.  Suppose  m  =  n. 
Under  this  hypothesis  we  shall  have 

d  . 

^  =  ^  =  00. 

This  result  implies  that  the  time  to  elapse  before  the  couriers 
are  together  is  greater  than  any  assignable  quantity,  or  infinity ; 
therefore  they  can  never  be  together. 

This  interpretation  is  in  accordance  with  the  conditions  of  the 
problem  under  the  present  hypothesis.  For,  at  12  o'clock  the 
couriers  were  d  miles  apart ;  and,  i?m  =  n,  they  were  traveling  at 
equal  rates.  Hence,  they  could  continue  to  travel  forever  without 
meeting. 

IV.  Suppose    d  =  0,  and  myn,  or  m<n. 

Then  X  =  — —  =  0. 

m  — 11 


132 


SIMPLE    EQUATIONS. 


That  is,  the  time  to  elapse  is  nothing.  This  result  implies 
that  the  couriers  were  together  at  12  o'clock,  and  at  no  other 
time. 

This  interpretation  is  confirmed  by  the  conditions  of  the  prob- 
lem. For,  if  d  =  0,  then,  at  12  o'clock,  B  must  have  been  with  A, 
at  the  point  P.  Moreover,  if  w  >  n,  or  m  <  w,  the  couriers  were 
traveling  at  difierent  rates,  and  must  have  been  either  approaching 
or  receding  from  each  other  at  all  times  except  at  the  moment  of 


^  =  0,    and    m  =  n. 


V.  Suppose 
Then 


0 
0' 


Here  the  value  of  ic  is  represented  by  one  of  the  symbols  of  in- 
determination.  This  result  implies  that  the  couriers  were  to- 
gether all  the  time. 

This  conclusion  is  evidently  confirmed  by  the  conditions  of 
the  problem.  For,  if  rf  =  0,  the  couriers  were  together  at  12 
o'clock ;  and,  if  m  =  n,  they  were  traveling  at  equal  rates,  and 
would  never  separate. 


236. 


SYNOPSIS    FOR    REVIEW. 


CHAP.  Yll.— Con. 
DISCUSSIONS. 


Teems  Used  . 


Interpretation  OP  f  Principle  1. 
Neg.  Results.      [  Principle  2. 

Zero. 

i  First  order. 
Second  order. 
Third  order,  etc. 
Infinity. 

Determinate  Quantity. 
Indeterminate  Quantity. 
I  Finite  Quantity. 

Symbols  of  indetermination.  i  x,  oo  x  n  5o  ,  oo  — oo 

0    A 


Symbols  of  zero, 
Problem  of  Couriers. 


I      A'  00 


CHAPTER    VIII. 

VANISHL\G  FRACTIOXS.-INDETERMIMTE   EQUATIONS  AW  PROS- 
LEMS.-IXCOMPATIBLE  EQUATIONS. 


VANISHING    FRACTJONS. 

227.  A  Vanishing  Fraction  is  one  which,  on  a  cer- 
tain supposition,  assumes  the  form   of  indetermination.    Thus, 

assumes  the  form  of  - ,  if  a;  =  «. 

X  —  a  0 

The  vahie  of  a  fraction  sometimes  reduces  to  the  form  of-,  for  a 

particular  supposition,  in  consequence  of  the  existence  of  a  factor 
common  to  both  terms,  which  factor  reduces  to  0  for  that  suppo- 
sition.   Thus,  the  fraction  tt^-t A  reduces  to  the  form  of  -, 

db  (a  —  b)  0 

if  fl  =  6,  because  the  factor  a  —  b  becomes  0  in  that  particular 

case.     But  if  this  factor  be  canceled,  and  the  supposition  that 

2 
a  =  ^  be  made  afterward,  the  value  of  the  fraction  will  be  7:. 

Before  deciding,  therefore,  upon  the  nature  of  the  symbol  -,  we 

must  ascertain  whether  it  results  from  a  factor  common  to  both 
terms,  which  reduces  to  0  for  the  supposition  made;  if  it  does 
not,  the  value  of  the  fraction  is  really  indeterminate. 


MULE. 

I.  Reduce  the  given  fraction  to  its  loioest  terms. 

II.  Malce  tlie  supposition  tuhich  would  cause  the  original  frac- 
tion to  assume  the  form  of  indetermination  ;  the  result  will  be  the 
value  of  the  fraction  for  that  supposition. 


134  INDETEBMINATE    EQUATIONS. 


EXAMPLES. 


1.  Find  the  value  of  —. — —^,  when  x  =  v. 

x^  —  f- 

Canceling  the  common  factor  x^  —  y%  we  have, 

which,  when  a;  =  y,  reduces  to  'Hy^', 

p^^  =  2y%  when  x  =  y. 
This  may  be  expressed  algebraically  as  follows : 

2.  Find  the  value  of   \  ^^(^-^)      \  ^^8.  14, 

1(1  -\-x){x  —  l)  I  «  =  !. 

3.  Find  the  value  of   1 1  ^^~  ^l^  I  Ans,  0. 

(3(a3— 62)  )  a  =  b. 

4.  Find  the  value  of   \  ~ J  >  Ans.  oo . 

5.  Find  the  value  of  (-, -r — = — rr-^ ^l  Ans.  oo. 

w  —  2aa:3  +  2a^x  —  ayx=a. 

6.  Find  the  value  of    ] x  7 — ^^^^-r^  i  Ans.  1. 

(a  —  x      (a  +  xy)  x  =  a. 


INDETERMINATE    EQUATIONS. 

228.  An  Indeterminate  Equation  is  one  in  which 
each  of  the  unknown  quantities  has  an  infinite  number  of  values. 

229.  A  single  equation  containing  tiuo  or  more  unknown 
quaiitities  is  indeterminate. 

Suppose  we  have  an  equation  containing  two  unknown  quan- 
tities, X  and  y,   for  example,  2a;  —  3?/ =  15.    For  every  value 


INDETERMIJSJ-ATE  EQUATIONS.  135 

which  we  please  to  ascribe  to  one  of  the  unknown  quantities  we 
can  determine  the  corresponding  value  of  the  other,  and  thus  find 
as  many  pairs  of  values  as  we  please  which  satisfy  the  given 
equation. 

Thus,  if  y  =  l,2,3,4:,6  .  .  .  .; 

then  x=%  lOJ,  12,  13^,  15  ...  . 

Again,  suppose  we  have  an  equation  containing  three  unknown 
quantities,  x,  y,  and-  z ;  for  example,  x  -{-  y  -{-  2z  =z  90.  For  every 
value  which  we  please  to  ascribe  to  two  of  the  unknown  quanti- 
ties we  can  determine  the  corresponding  value  of  the  third,  and 
thus  find  as  many  sets  of  values  as  we  please  which  satisfy  the 
given  equation. 

Thus  if  i'^^^'  2,  3,  4,  5 , 

^^"""^'^  13^  =  0,1,2,5,8....; 

then  X  =  88,  85,  82,  77,  72  ...  . 

A  similar  course  of  reasoning  is  applicable  to  an  equation  con- 
taining more  than  three  unknown  quantities. 

330.  Equations  are  indeterminate  if  the  number  of  unknown 
quantities  involved  exceeds  the  number  of  equations. 

For,  by  eliminating,  we  can  obtain  a  single  equation  contain- 
ing two  or  more  unknown  quantities,  which  is  indeterminate 
(239). 

Thus,  suppose  we  have  the  two  equations 

x^    y-{-2z=    90     .     .     .     (1), 

6x -\- 2y  —  2z  =  366    .     .     .     (2). 

Eliminating  z, 

6a; +  3?/ =  456    .    .    .     (3), 

which  is  indeterminate. 

231.  An  equation  containing  only  one  unhnoion  quantity 
may  he  indeterminate  in  consequence  of  certain  relations  which 
subsist  betiveen  the  known  quantities. 


136 


INDETERMINATE    EQUATIONS. 


If  we  solve  the  equation 

ax  -[-  b  =  ex  -\-  d 
d-b 


we  obtain 


X  = 


a  —  c 

Now,  if  d  =  b,  and  a  =  c, 

0 
^=0    •    • 


(1), 
(2). 


(3); 


hence,  under  this  hypothesis,  the  value  of  x  is  indetenninate. 

But,  ]£  d  =  b,  and  a  =  c,  (1)  becomes 

ex  -\-  b  =  ex  -{-  b    .     .    .     (4), 

which  is  an  identity,  and  may  therefore  be  satisfied  for  any  value 
of  X  (178). 

Here,  then,  we  have  one  unknown  quantity  and  7io  equation ; 
that  is,  no  equation  of  condition  (179). 

232.  Two  equations  involving  tivo  unknown  quantities  may 
be  indeterminate  in  consequence  of  certain  relations  which  sub  list 
among  the  known  quantities. 


(1), 

(2), 
(3), 

and  V  =  "^ — ^    .    .     .     (4). 

(5), 
and  bcz=  ad (6), 


If  we  solve  the 

equations 

ax  -\-by  =  r 

ex  -\-  dy  =  s 

obtain 

dr  —  bs 
~ad—bc 

as  —  cr 

L 

y~  ad-bc 

Now,  if 

dr  =  bs  .    . 

I 

be  =  ad  .    . 

then,  by  multiplying  (5)  by  (6),  member  by  member,  and    re- 
ducing, 

cr  =  as (7) ; 

.'.    (3)  and  (4)  become 


INDETERMIiq^ATE  PEOBLEMS.  137 

0      ,       0 

x  =  -,    and    y  =  -. 

Let  us  now  see  what  is  implied  by  the  relations  (5)  and  (6). 

From    (5)    we   have    d  =  —,    and    from    (6),    c  =  ^  =  —. 

These  values  of  d  and  c  reduce  (2)  to  (1),  and  we  then  have  only 
one  equation  containing  two  unknown  quantities,  which  is  inde- 
terminate. 

Cor.. — The  four  theorems  which  have  just  been  demonstrated 
may  be  reduced  to  the  following  one : 

Indetermination  arises  if  the  number  of  unTcnown  quantities 
exceeds  the  number  of  equations. 

INDETERMINATE    PROBLEMS. 

233.  An  Indeterminate  JProbleni  is  one  which  ad- 
mits of  an  infinite  number  of  solutions. 

We  may  often  limit  the  number  of  solutions  by  imposing  the 
condition  that  the  values  of  the  unknown  quantities  shall  be  pos- 
itive integers. 

When  an  indeterminate  problem  is  expressed  in  algebraic  lan- 
guage, it  will  be  found  that  the  number  of  unknown  quantities 
exceeds  the  number  of  equations. 

PnOBZEMS. 

1.  A  boy  paid  50  cents  for  some  apples  and  oranges,  giving  2 
cents  each  for  apples  and  10  cents  each  for  oranges.  How  many 
of  each  did  he  buy  ? 

Let  X  =  the  number  of  apples,  and  y  =  the  number  of  oranges; 
then,  by  the  question, 

2x-\-  10?/ =  50; 

whence,  a;  =  25  —  5y. 

Now,  if  x  and  y  are  to  be  positive  integers,  y  must  be  some  in- 
teger between  0  and  5.    Let 

y=    1,     2,     3,  4; 

then  X  =  20,  15,  10,  5. 


138  INDETERMINATE  PROBLEMS. 

2.  Find  two  positive  integers  sucli  that  12  times  the  one  ex- 
ceeds 13  times  the  other  by  9. 

Let  X  =  one  of  the  numbers, 

and  y  =  the  other ; 

then,  by  the  question, 

12a;  — 13^  =  9; 

whence,  x=     ^^^     =^  +  S^' 

t/  +  9 
Since  x  and  y  are  to  be  positive  integers,  ^  must    b^ 

an  integer.    Let 

y  +  9 

whence^  ^  =  12?i  —  9. 

r   .  1   o  o  .  ^i,        ( 2^  =  3,  15,  27,  39 

Let  ..  =  1,  2,  3,4....;  then  1^^^^^^;  3^^  ^3^^^^ 

3.  A  man  bought  100  animals  for  $100 ;  sheep  at  $3^  each^ 
calves  at  $1^,  and  pigs  at  $j^.  How  many  did  he  buy  of  each  kind  ? 

Let  X  =  the  number  of  sheep, 

y  =  the  number  of  calves, 
z  =  the  number  of  pigs ; 

then,  by  the  question, 

x+     y+    z  =  100    .    ,    .     (1), 

3^2;  +  li2/ +  i^  =  100    .    .    .     (2). 

Combining  (1)  and  (2),  eliminating  z, 

18a;  +  5y  =  300    ...     (3); 

whence,  y=60 —    .    .    .     (4). 

o 

From  (1)  it  is  evident,  that  if  x  and  y  are  positive  integers 

whose  sum  is  less  than  100,  z  will  be  a  positive  integer  also. 

From  (4)  it  is  e\ddent  that  x  must  be  a  multiple  of  5,  and  that 

18a; 

—^  must  be  less  than  60. 
o 


INCOMPATIBLE    EQUATIOKS.  139 

Let  X  =:   5,  10,  15 ; 

then  y  =  42,  24,     6, 

and  z  =  53,  66,  79. 

4.  The  sum  of  three  positive  integers  is  11 ;  and  if  the  first  be 
multiplied  by  3,  the  second  by  5,  and  the  third  by  7,  the  sum  of 
the  products  will  be  57.    What  are  the  numbers  ? 

I  a;  =  4,  3,  2,  1, 

Ans.   i  .y  =  2,  4,  6,  8, 

('z=6,  4,  3,  2. 

5.  Divide  200  into  two  parts,  such  that  if  one  of  them  be  di- 
vided by  6  and  the  other  by  11,  the  respective  remainders  may  be 
5  and  4.  Ans,  185,  15 ;  119,  81 ;  53, 147. 

6.  Can  the  equation  4a;  +  6y  =  27  be  solved  in  positive  in- 
tegers ? 

7.  Find  the  least  number  which,  being  divided  by  5,  leaves  a 
remainder  3,  and  divided  by  7  leaves  a  remainder  5.      A^is.  33. 

8.  Solve  the  equation   8a;  -f  13?/  =  159    in  positive  integers. 

.         j  a;  =  15,  2. 
Ans.    i         _    ^. 
(y  =  3y  11. 

INCOMPATIBLE    EQUATIONS. 

234.  Incompatible  liquations  are  those  which  can- 
not be  satisfied  for  the  same  values  of  the  unknown  quantities. 

235.  Equations  are  said  to  be  Independent  when  they 
express  conditions  essentially  different,  and  Dependent  when 
they  express  the  same  conditions  under  different  forms. 

Thus,    ]  K    _  qq  f    ^^®  independent  equations. 

But       \  ^        J  ~  ^^c    are  dependent  equations,  since  the 
(  2a;  +  6i/  =  38  ) 

second  may  be  obtained  from  the  first  by  multiplying  both  mem- 
bers by  2. 

236.  If  the  number  of  independent  equations  exceeds  the 
number  of  unknown  quantities,  these  equations  may  be  incom- 
patible. 


140  INCOMPATIBLE    EQUATIONS. 

Let  US  consider  the  three  equations 

rr  +  y  =  8    .    .     .     (1), 
x-y=:2    .    .     .     (2), 

?=2     .     .     .     (3). 

y 

Combining  (1)  and  (2),  we  find  a;  =  5,  and  y  =  3 ;  but  these 
values  will  not  satisfy  (3).  In  like  manner  it  may  be  shown  that 
the  values  which  satisfy  (2)  and  (3)  do  not  satisfy  (1),  and  that 
the  values  which  satisfy  (1)  and  (3)  do  not  satisfy  (2). 

237.  If  the  number  of  independent  equations  exceeds  the 
number  of  tinknoiun  quantities,  such  relations  between  the  hnotun 
quantities  can  be  found  as  icill  make  the  equations  compatible. 

Let  us  consider  the  equations 


x^y  =  s 

.    .    . 

•  (1), 

x  —  y  =  d 

.    .    . 

(2). 

x  =  ay.    .    . 

(3). 

Combining  (1) 

and  (2),  we  find 

x  = 

s  -\-  d 
2     ' 

y  = 

s-d 
2     • 

Substituting  these  values  in 

(3), 

s  +  d 
2      ~ 

'H- 

'-}■■ 

ince, 

s  +  d 
''  =  s-d 

.    .    . 

(4). 

If  the  relation  expressed  by  (4)  subsists,  (1),  (2),  and  (3)  will 
be  compatible.     Thus,  the  equations, 

x  +  y  =  9, 
x  —  yz=zd, 

x  =  2y, 
are  compatible,  for 

94-3 


2  = 


3* 


SYNOPSIS    FOK    KEVIEW. 


141 


Cor. — In  order  that  a  problem  may  be  determinate,  the  con- 
ditions must  furnish  as  many  independent  equations  as  there  are 
unknown  quantities. 

ScH.  1. — When  a  problem  contains  more  conditions  than  are 
necessary  for  determining  the  values  of  the  unknown  quantities, 
those  that  are  unnecessary  are  termed  redundant. 

ScH.  2. — A  problem,  from  which  incompatible  equations  are 
deduced,  is  called  an  impossible  jwohlem.  Such  a  problem  is  said 
to  involve  incompatible  conditions. 


338. 


SYNOPSIS    FOR    REVIEW. 


CHAPTER  Vin. 


'  Vanishing  Fractions.    Investigation.    Rule. 
229. 


Theorems  relat- 
ing TO  Indetermi- 
nate Equations. 


230, 
231. 
232. 

Reduction  of  the  four  theorems 

to  ONE. 

Indeterminate  Problems.   No.  solutions  limited,  how. 

i  Terms  j  Dependent  Equations. 
used.   I  Independent  Equations. 
^  *>36 


Theorems.  \  ' 


237.  Cor.;8ch.l,!2. 


CHAPTER   IX, 
INEQUALITIES. 


239.  An  Inequality  consists  of  two  expressions  con- 
nected by  the  sign  of  inequality. 

The  First  Member  of  an  inequality  is  the  expression  on  the 
left  of  the  sign  of  inequality,  and  the  Second  Member  is  the  ex- 
pression on  the  right  of  the  sign. 

240.  Two  inequahties  subsist  in  the  same  sense  when  the 
first  member  is  the  greater  in  each,  or  the  less  in  each.  Thus, 
5  >  3  and  7  >  4  subsist  in  the  same  sense. 

Two  inequahties  subsist  in  a  contrary  sense  when  the  first 
member  is  the  greater  in  one,  and  the  less  in  the  other.  Thus, 
5  >  1  and  4  <  8  subsist  in  a  contrary  sense. 

241.  If  the  same  quantity  be  added  to,  or  subtracted  from, 
each  member  of  an  inequality,  the  resulting  inequality  will  subsist 
in  the  sams  sense. 

For,  suppose  a  >  J ;  then  a  —  J  is  positive  (118).  Again, 
since  a  ±  <^  —  (&  ±  c)  =  a  -—  J,  it  follows  that  «  ±  c  —  (J  ±  ^) 
is  positive ; 

Cob.  1. — The  rule  for  the  transposition  of  terms  in  equations 
is  applicable  to  inequalities.    Thus,  if 

then  a3  +  £2  _  ^ab  >  )i>ab  —  2ab  +  c^, 

or  a2--2a&  +  ^>c8. 


INEQUALITIES.  143 

Cor.  2. — If  an  equation  be  added  to  an  inequality,  member  to 
member,  or  subtracted  from  it,  member  from  member,  the  result- 
ing inequality  will  subsist  in  the  same  sense.    Thus,  if 

a>h, 

and  ^  =  2^> 

then  «  ±  ^  >  ^  ±  y« 

242.  If  an  inequality  he  subtracted  from  an  equation^  mem- 
her  from  member,  the  sign  of  inequality  will  be  reversed. 

For,  suppose  ^  =  y, 

and  «  >  ^ ; 

then  x  —  a  —  {y  —  b)z=h  —  a, 

and  5  —  a  is  negative ; 

x  —  a<^y  —  h, 

243.  If  both  members  of  an  imquality  be  multiplied  or  di- 
vided by  the  same  positive  quantity,  the  resulting  inequality  will 
subsist  in  the  same  sense. 

For,  suppose  m  to  be  positive,  and 

a>h] 
then,  since  a  —  J  is  positive,  m{a  —  h)   and  —  (a  —  J)  are  pos- 
itive ; 

maymb    and    —  >  — 
m      m 

244.  If  both  members  of  an  inequality  be  multiplied  or  di- 
vided by  the  saine  negative  quantity,  the  resulting  inequality  ivill 
subsist  in  a  contrary  sense. 

i  or,  suppose  m  to  be  negative,  and 

a>b', 
then,  smce  a  —  &  is  positive,  m(a  —  b)  and  —  (flf  —  J)   are  neg- 
ative ; 

ma<,mb    and    —  <  — 
m      7n 


144  INEQUALITIES. 

Cor. — If  the  signs  of  all  the  terms  of  an  inequality  be  changed, 
the  sign  of  inequality  must  be  reversed.  For  changing  the  signs 
of  all  the  terms  is  equivalent  to  multiplying  each  member  by  —  1. 

245.  If  tivo  or  more  inequalities  subsisting  in  the  same  seiise 
he  added,  member  to  member,  the  resulting  inequality  luill  subsist 
in  the  same  sense  as  the  given  inequalities. 

For,  if  ay  by    a' y  b',    and    a"  >  b", 

then     a~-bi     a'  —  b\     and    a"  —  b"    are  positive ;  therefore, 

a  —  b  -^  a'  —  b'  -\-  a"  —  b'\      or      a  +  a'  +  a!'  —  (^  +  Z>'  +  b"), 

is  positive ; 

a  +  a' +  a"  >  Z>  +  J' +  J". 

ScH. — If  one  inequality  be  subtracted  from  another  subsisting 
in  the  same  sense,  the  result  will  not  always  be  an  inequality  sub- 
sisting in  the  same  sense  as  the  given  inequalities,  or  an  inequality 
ai  all. 

Take  the  two  inequalities 

4<7, 
and  2  <  3. 

By  subtraction,  2  <  4. 

Here  the  result  is  an  inequality  subsisting  in  the  same  sense 
as  the  given  inequalities. 

But  take  9  <  10, 

and  6  <   8. 

By  subtraction,  3  >    2. 

Here  the  result  is  an  inequality  subsisting  in  a  sense  cont'^ary 
to  that  of  the  given  inequalities. 

Again,  take  9  <  10, 

and  6<    7. 

By  subtraction,  3  =   S.- 

Here the  result  is  an  equation. 


SOLUTIOIS^.  145 

346.  TJie  Solution  of  an  inequality  consists  in  trans- 
forming it  in  such  a  manner  that  the  unknown  quantity  may 
stand  alone  as  one  of  its  members.  The  other  member  will  then 
denote  one  li7n,it  of  the  unknown  quantity. 

EXAMPLES, 

Solve  the  following  inequalities  : 
,  X  ,   2x  ^*^x  ,   9 

Multiplying  both  members  by  20, 

10a;4- 8a;>15a;'4-45; 
transposing  and  reducing, 


3a;>45; 

ence, 

a;>15. 

2.     5:.>y+14. 

u47zs.  a;  >  4. 

2x      2x      2x 

^715.  a;  <  3. 

8  ^  4  ^  6  ^  12 

^WSv^<  14:. 

247.  If  there  be  given  an  inequality  and  an  equation,  involv- 
ing two  unknown  quantities,  a  limit  of  each  unknown  quantity 
may  be  found  by  ehmination. 

EXAMPLES. 

1.  Find  a  limit  for  x  and  y  in  the  following  groups: 


(2a;  +  5«^>16    .     . 
(2a;  +  2^  =  12      .    . 

.    (1), 
.    (2). 

Subtracting  (2)  from  (1), 

4y>4j 

whence,                                   y  >  1. 

10 

146 


INEQUALITIES. 


If  we  substitute  1  for  y  in  (2),  the  first  member  will  be  made 
less  than  the  second ;  hence, 

2rr  +  l<12; 

whence,  a;  <  oj* 


j2a;  +  4y>30) 
(3:r  +  2^  =  31) 

(5a:-3?/<15) 
(9a;-f-2«/  =  46) 

(4a;+    5^  =  68)' 

(5a:  +  3y>121) 
(7a;+4y  =  168i' 


6.    . 


8  6      -^ 

3£--24      rr~2/_-,o 


-4ws.  a;  <  8,  2/  >  ^i* 

Ans.  x<^^,y>  2f|. 

-47^5.  a;  <  13,  yy  ^. 

Ans.  a:  <  20,  y  >  7. 


248. 


CHAPTER  IX. 
mEQUALITIES. 


SYNOPSIS    FOR    REVIEW. 

Membeks. 

stjbsisting  in  the  same  sense. 

Subsisting  in  a  contrary  sense. 

241 Cor.  1,  8» 

242. 

Theorems  ....  J  243. 

244.— C<7r. 
,  245.— .a^. 

SOLtTTION. 

Combination  of  an  Equation  with  an  Inequality. 


OHAPTEE   X. 
Il^YOLUTION   A]^D   EYOLUTIOJSr. 


INVOLUTION. 


249.  A  Potver  of  a  quantity  is  the  product  of  factors  each 
of  which  is  equal  to  that  quantity.  A  quantity  is  said  to  be  raised 
or  involved  when  any  power  of  it  is  found. 

250.  Involution  is  the  process  of  raising  a  given  quantity 
to  any  required  power. 

251.  The  Base  or  Hoot  of  a  power  is  one  of  the  equal 
factors  of  the  power,  and  the  Degree  of  a  power  is  equal  to  the 
number  of  times  the  base  is  used  as  a  factor  to  produce  the  power. 
Thus,  a^  is  the  third  power  or  cube  of  a,  and  a  is  the  base  of  a\ 

252.  The  Exponent  of  the  Potver  is  the  exponent 
which  indicates  the  power  to  which  the  given  quantity  is  to 
be  raised. 

253.  A  Perfect  Power  of  the  n^^  degree  is  a  quantity 
which  can  be  resolved  into  n  equal  factors.  Thus,  a^  —  "Zab  +  W' 
is  a  perfect  power  of  the  second  degree. 

254.  An  Imperfect  Power  of  the  n*^  degree  is  a  quan- 
tity which  cannot  be  resolved  into  n  equal  factors.  Thus,  a^—  h^ 
is  an  imperfect  power. 

A  quantity  may  be  a  perfect  power  of  one  degree,  but  an  im- 
perfect power  of  another  degree.  Thus,  a^  —  2al)  +  ^^  jg  ^  perfect 
power  of  the  second  degree,  but  an  imperfect  power  of  the  third 
degree. 


148  INVOLUTION". 

255.  Tlie  Sign  of  the  Power, — Any  power  of  a  posi- 
tive quantity  is  positive ;  for,  when  all  the  factors  are  positive,  the 
product  is  positive.  If  the  quantity  to  be  involved  is  negative, 
the  even  powers  will  be  positive,  and  the  odd  powers  will  be  neg- 
ative. 

For,  {  —  a){—a)  =  a\  (—  a)  (—  a)  (—  a)  —a^  (— «)=  —a\ 
(—«)(—«)(—«)(—«)=  —  a^  (—  a)  =  a\  and  so  on. 

256.  The  7i*^  JPotver  of  a  Product,— It  follows,  from 
Art.  249,  that  {abY=ab  xdbxdb  .  .  .U)  n  factors=a  xaxa.  . . 
to  n  factors  xoxbxb  . . .  to  n  factors  =  a"b\  In  like  manner, 
{abcY  =  a^'b^c",  and  {abc  .  . .  ^•)'*  =  a"Z>"c"  .  ,  ,  k". 

. ' .     The  71^  power  ofjhe  product  of  any  number  of  factors  is 
equal  to  the  product  of  the  n^^  poivers  of  those  factors 

Again,         (a^^f  =  a"  x  a"  =  a"+"  =  a^, 

(a")^  =  «•»  X  a"  X  a"  =  «"+"+"  =  a**, 

.  • .  If  the  n^  power  of  a  quantity  be  raised  to  the  m^  poioer, 
the  result  will  be  the  mn*^  power  of  tMt  quantity. 

257.  Hie  Coefficient  of  the  Power.— Since  (5aY  = 
5a  X  ba  X  6a  =  bhi^  =  12oa%  and  (5a)"  =  S^a",  it  follows  that 

The  coefficient  of  the  n^  power  of  a  quantity  is  the  n*^  power 
of  the  coefficieiit  of  that  quantity. 

258.  To  find  any  power  of  a  monomial. 

B,  ULE. 

Raise  the  numerical  coefficient  to  the  required  power,  and  write 
after  the  result  all  the  letters  of  the  given  monomial,  giving  to  each 
an  exponent  equal  to  the  product  of  its  original  exponent  by  the 
exponent  of  the  power, 

1.  Find  the  3d  power  of  2a^^c.  Ans,  %a%^<^. 

2.  Find  the  6th  power  of  the  5th  power  of  a%(^. 

Ans,  a^b^(^. 


I]S"VOLUTION. 


149 


3.  Find  the  5th  power  of  —  abx"".  Ans.  —a^^a^. 

4.  Find  the  4th  power  of  —  aWc^.  Ans.  a%^\hl^. 

5.  Find  the  5th  power  of  ^a^x\  Ans.  243«%io. 

6.  Find  the  7th  power  of  aWc  (—  x^fzf^). 

Ans.  —  a^Wc^ x^yh^. 

7.  Find  the  3d  power  of  (—  aMc^)  (—  a^bh).       Ans.  aWc^ 

8.  Find  the  5th  power  of  (—  af  (—  by  (—  cy. 

Ans.  —aW^c^. 

9.  Find  the  m^  power  of  —  a^b^(^  when  m  is  an  even  positive 
integer.  Ans.  a^b^'^c^"'. 

10.  Find  the  m^^  power  of  —  (abcY  when  m  is  a  positive  in- 
teger. A71S.  ±  c^'^b^c^K 

359,  To  find  any  power  of  a  polynomial. 


RULE, 


Find  the  product  of  as  many  factors,  each  of  ichich  is  equal 
to  the  given  polynomial,  as  there  are  units  in  the  exponent  of  the 
required  power. 


11,1.  USTBATIONS. 


a  -\-b 
a  -\-b 
a^  +  ab 

-j-ab-hb^ 


a^+2ab  +b^ 

a  -\-b 

a^+2a^-\-  ab^ 
-\-a^   -^2ab^  +  b^ 


a^  +  Za%  +  ^alP  +^3 
a  -\-b 

+  a^b  +  ^aW  +  dab^+¥ 


a2  +  2ab  +  ^^,       a^-{-  Sa^  +  dab^  +  b%      a'^  +  ^a^  +  Qa^^  +  ^ab^  +  b\ 

{a-\-bY  =  a^-\-  2ab-^b% 

(a  +  by  =  a3  +  Sa^  +  3«^^  +  b% 

(a  +  by  =  a^  +  4:a^b  +  6^2^,2  4.  4a§3  ^  j4. 

In  like  manner  it  may  be  shown  that 

(a  —  by  =  «2  -  2a,b  +  b\ 

(a  —  by  =  a^  —  M%  +  3fl^2  _  js^ 

\a  —  by  =  a^  —  ia^b  +  QaW  —  4ab^  +  i^- 


150  INVOLUTION 

Cor.  1. — Since  [a^'Y  =  (a")"*  (256),  we  may  reach  the  same 
result  by  different  processes  of  involution.  For  example,  we  may 
find  the  sixth  power  of  a  -\-  ihj  repeated  multiplication  by  a  +  Z>; 
or  we  may  first  find  the  cube  of  a  -\-  b,  and  then  square  the  re- 
sult ;  or  we  may  first  find  the  square  of  a  -\-  b,  and  then  cube  the 
result. 

The  work  may  sometimes  be  abridged  by  using  the  principle 
expressed  by  the  equation  «*"«"= «»'*+".  Thus,  we  may  find  the 
fifth  power  of  a  -{-  b  by  multiplying  the  cube  of  a  +  b  hy  the 
square  of  a  +  b. 

Cor.  2. — It  may  be  shown  by  actual  multiphcation,  that 
(a-{-b-^cy=a^+2a(b  +  c)-\-b^+2bc-^c^,    and 
(a  +  b-\-c-^dy=a^-\-2a{b-\-c  +  d)  +  b^-\-2b{c-{-d)-i-ci-{-2cd+(P. 

Hence,  we  may  infer  that 

77ie  square  of  any  polynomial  is  equal  to  the  sum  of  the 
squares  of  its  terms,  together  with  twice  the  sum  of  the  products 
obtained  by  multiplying  each  term  by  the  sum  of  all  the  terms 
which  follow  it, 

EXAMPLES. 

1.  Find  the  square  of  {a  —  b-\-  c). 

Ans.  a^  +  l^  -\-  c^  —  2ab  -\-  2ac  —  2bc, 

2.  Find  the  square  of  \  -\-  x  -\-  ofi  -^  a^. 

Ans.  1  +  2:r  +  3a:2  +  4a;3  ^  3^  _^  2:^5  +  2:6. 

3.  Find  (1  -  2a;  +  ^3?Y. 

Ans.  1  —  4a;  +  lOx^  —  12a:3  +  ^x^. 

4.  Find  {a -\- b  —  cf. 

Ans.  a^  +  b^—(^+^a\b—c)  +  W{a-c)  +  d(^{a  +  b)^Qabc. 

5.  Find  (1  +  22;  +  x^f. 

Ans.  1  +  6a;  +  15a;2  +  20a;3  4.  15^4  _f_  6:^5  +  a;^. 

6.  Find  {a  +  bf. 

Ans.  a^  +  Wb  +  Iba^lr^ + 20^3^ + lha^¥  4-  Qa¥ + ¥. 


(I)" 


EVOLUTION".  151 

260.  To  find  any  power  of  a  fraction. 

a  X  a  X  a  ...  to  n  factors      a" 


X  T  X  T  to  71  factors  = 


b       b       b  ~bxbxb...tx)n  factors  ~~  6"' 

Hence, 

The  n^^  power  of  a  fraction  is  a  fraction  tuhose  numerator  is 
the  n^^  poiver  of  the  given  numerator,  and  whose  denominator  is\ 
the  n^  power  of  the  given  denominator, 

EXAMPZES. 


W  -  ^/  • 


3.  Find    ^^-^-r   •  Ans. 


a^—'^ab-\-¥ 


4   Find  I   ■  . .  ^^,.o   

5.   Show  that  ( ^^ )  +i ._2J_.J=J.. 

EVOLUTION. 

261.  Let  n  be  any  positive  integer ;  then 

The  n*^  Hoot  of  any  given  quantity  is  a  quantity  the  n*^ 
power  of  which  is  equal  to  the  given  quantity. 

262.  Evolution^  or  T/ic  Extraction  of  Hoots,  is 

the  process  of  finding  any  root  of  a  given  quantity.    Evolution  is 
the  converse  of  involution. 

263.  Tlie  Sif/n  of  the  Boot— If  the  i7idcx  of  the  root 
to  be  extracted  be  an  odd  number,  the  sign  of  the  root  will  be  the 
same  as  the  sign  of  the  given  quantity  (255).  Thus,  V«^  =  ay 
and  \/—a^=  —  a. 


152  EVOLUTION. 

If  the  index  of  the  root  to  be  extracted  he  an  even  number, 
and  the  given  quantity  be  positive,  the  root  may  be  either  positive 
or  negative.    Thus,  ^/a^  =  ±  a. 

264.  If  the  index  of  the  root  to  be  extracted  be  an  even  num- 
ber, and  the  given  quantity  be  negative,  the  root  cannot  be  ex- 
tracted ;  because  no  quantity  raised  to  an  even  power  can  produce 
a  negative  result  {2>55).  The  indicated  even  root  of  a  negative 
quantity  is  caUed  an  Imaginary  Quantity,  Thus,  V— 9, 
V— «^,  and  V—  {(I  +  if  are  imaginary  quantities. 

365.  To  find  any  root  of  a  monomial. 
(3a'»Z''-)"  =  S^a"^^  (358); 

^3«^m«^rn  _  ^^m^r    (261). 

RULE. 

Extract  the  required  root  of  the  numerical  coefficient,  and  write 
after  the  result  all  the  letters  of  the  given  monomial,  giving  to  each 
an  exponent  equal  to  the  quotient  obtained  hy  dividing  its  original 
exponent  hy  the  index  of  the  root, 

1.  Find  VSo^^^-  -4w5.  2a^l^c, 

2.  Find  V^^^^-  ^ns.  ±  aH^c^^ 

3.  Find  V— «^^^-  ^^^s.  —  ahxT. 
4  Find  \/(^bVd^  Ans.  ±  aWcdK 


5.  Find  a/«*"^^"^"*'*>  when  m  is  an  even  positive  integer. 

Alls,  ih  a%^(^. 

6.  Find  ^  —  a''¥c^, 

266.  To  find  the  square  root  of  a  polynomial. 

Since  the  square  root  of  c?  +  lal  +  J^  ig  a  -^-l,  we  may  be 
led  to  a  general  rule  for  the  extraction  of  the  square  root  of  a 


EVOLUTION.  153 

polynomial  by  observing  in  what  manner  a  -{-  b  may  be  derived 
from  a2  _|_  2ab  +  bl 


2a-i-b\2ab  +  b^ 


Arrange  the  terms  according  to  the  descending  powers  of  a ; 
then  the  square  root  of  the  first  term,  a%  is  a,  which  is  the  first 
term  of  the  required  root.  Subtracting  its  square,  that  is,  a^^ 
from  the  given  polynomial,  we  obtain  the  remainder  2ab  -\-  bK 
Dividing  2ab  by  2a,  we  obtain  b,  which  is  the  second  term  of  the 
root.  Multiplying  2a  -{-  bhj  b,  and  subtracting  the  product  from 
the  first  remainder,  we  obtain  0  for  the  second  remainder ;  hence 
a  +  b  is  the  required  root. 

When  the  root  contains  three  or  more  terms,  it  may  be  found 
by  a  similar  process.    Thus, 

fl2  +  2a5  +  ^  +  2  (flj  +  J)  c  +  c2 1 «  +  5  +  c 


2a-{-b\2ab  +  b^ 
'  2ab  +  b^ 


2(a-{-b)  +  c\2{a-{-b)c-^c^ 
^2{a  +  b)c  +  c^ 


The  first  term  of  the  required  root  is  a.  Subtracting  a^  from 
the  given  polynomial,  we  obtain  the  remainder  2ab  +  b^-{-2(a-^b)c 
+  c2.  Dividing  the  first  tenn  of  this  remainder  by  2a,  we  obtain 
the  second  term  of  the  root.  Multiplying  2a  -\- b  by  b,  and  sub- 
tracting the  product  from  the  first  remainder,  we  obtain  the  sec- 
ond remainder,  2  {a  -\- b)  c  -{-  c^.  Dividing  the  first  term  of  the 
second  remainder  by  2a,  we  obtain  tiie  third  term  of  the  root. 
Multiplying  2  (a  +  Z>)  +  c  by  c,  and  subtracting  the  product  from 
the  second  remainder,  we  obtain  0  for  the  third  remainder ;  hence 
a  -{-b  +  c  is  the  required  root. 

We  call  2a  the  partial  divisor,  2a  -\-b  the  first  complete  divi- 
sor, and  2  (a  +  2>)  +  c  the  second  complete  divisor. 


154  EVOLUTION". 

RULE, 

I.  Arrange  the  given  polynomial  according  to  the  powers  of 
one  of  its  letters. 

IL  Extract  the  square  root  of  the  first  term;  the  result  will  he 
the  first  term  of  the  required  root.  Subtract  the  square  of  this 
term  from  the  given  polynomial. 

III.  Divide  the  first  term  of  the  remainder  by  twice  the  first 
term  of  the  root,  and  annex  the  result  to  the  first  term  of  the  root 
and  also  to  the  divisor  ;  then  multiply  the  divisor  thus  completed 
by  the  second  term  of  the  root,  and  subtract  the  product  from  the 
first  remainder. 

IV.  Take  twice  the  sum  of  the  first  and  second  terms  of  the 
root  for  a  second  divisor.  Divide  the  first  term  of  the  second  re- 
mainder by  the  first  term  of  the  second  divisor,  and  annex  the  re- 
sult to  the  part  of  the  root  already  found  and  also  to  the  second 
divisor ;  then  multiply  the  divisor  thus  completed  by  the  third 
term  of  the  root,  and  subtract  the  product  from  the  second  re- 
mainder, 

V.  If  the  required  root  contains  additional  terms,  proceed  in 
like  manner  until  all  the  terms  are  found. 

Cor.  1. — If  the  first  term  of  the  arranged  polynomial  is  not  a 
perfect  square,  or  if  the  first  term  of  any  arranged  remainder  is 
not  divisible  by  twice  the  first  term  of  the  root,  the  exact  square 
root  cannot  be  found. 

Cor.  2. — All  even  roots  admit  of  the  double  sign  (263) ; 
hence  the  square  root  of  a^  +  2ab  +  J^  jg  _  (^j  _^  j)^  ag  ^rgu  j^g 
a  -\-b.  In  fact,  the  first  term  in  the  root,  which  we  found  by 
extracting  the  square  root  of  a\  might  have  been  —  a ;  and  by 
using  this  we  should  have  obtained  —  b  for  the  second  term  of 
the  root  * 

EXAMPLES 

Find  the  square  root  of  each  of  the  following  expressions : 

1.  ^x^  —  123^  +  hx^  +  6a;  -f  1.         Ans.  ±  {2x^  —  3a;  —  1). 

2.  1  +  42:  -f  10a;2  +  127?  +  ^x^.       Ans.   ±  (1  +  2a;  +  ox^). 


EVOLUTION.  155 

3.  9x^  +  12a;3  _|_  22a:2  _|_  i2a;  4.  9.     A71S.  ±  {Sx^  +  2a;  +  3). 

4.  9^2  -j-  I2ah  +  4^  4-  6ac  +  4^>c  +  c^ 

Ans.  ±  (3a  +  25  +  c). 

5.  4^4  —  12a3  4-  25a2  _  24«  +  16.     Ans.  ±  (2^2—  Za  +  4). 

6.  I62:*  —  Uabx^  +  16^2^2  _|_  4^2^2  _  g^js  ^  4^4^ 

7.  a;(5  _  4^  _|_  i0a;4  _  20a;3  +  250:2  —  24.2;  +  16. 

8.  Slx^  —  432a;3  ^  864a:2  —  7G8a;  +  256. 

9.  {a  -by -2  (a2  4.  ^)  (^  _  j)2  +  2  (a*  +  ^4). 

10.    a^^M  +  (^-{-d^  —  2«2  (Z,2  ^  J2)_2J2(^_^2)_,_2c2(a2_^2). 

267.  Wlien  a  Trinomial  is  a  JPerfect  Square. — 

Since  {a  ±  hY  =z  a^  +  %ah  +  h^.  it  follows  that  a  ainomial  is  a 
perfect  square,  if  two  of  its  terms  are  squares,  and  the  other  term 
is  twice  the  product  of  the  square  roots  of  these  two. 

When  a  trinomial  is  a  perfect  square,  its  square  root  may  be 
found  by  extracting  the  square  root  of  each  of  the  square  terms 
and  connecting  the  results  by  the  sign  of  the  other  term. 

1.  Extract  the  square  root  of  \.x^  —  Vtxy  +  ^y\ 

This  is  a  perfect  square,  because  4a:2  and  9?/2  are  squares,  and 
Vlxy  is  equal  to  twice  the  product  of  the  square  roots  of  these 
terms.  The  square  root  of  4a;2  is  2a;,  and  the  square  root  of  9^2  jg 
Zy.  Connecting  these  results  by  the  sign  of  the  term  12a:?/,  we 
obtain  2a;  —  3y,  or  Zy  —  2x. 

2.  Extract  the  square  root  of  a;2  _|_  ^xy  +  9^2, 

Ans.  ±  (a;  4-  Zy). 

3.  Extract  the  square  root  of  9^2  4-  25^>2  _  30fl;5. 

Ans.  ±  (3a  —  5Z>). 

4.  Extract  the  square  root  of  9a!2  _  Vlab  4-  16^2. 

268.  An  expression  wJiich,  in  its  simplest  form,  is  a  bino- 
mial, cannot  be  a  perfect  square.  For  the  square  of  a  monomial  is 
a  monomial,  and  the  square  of  any  polynomial  contains  at  least 
three  terms. 


156  EVOLUTION. 

269.  To  find  the  square  root  of  a  number. 

Pkinciples. — 1.  Tlie  square  of  a  number  consisting  of  tens 
and  units  is  equal  to  the  sum  of  the  squares  of  the  tens  and  the 
units  increased  hy  tiuice  their  prod^ict.    Thus, 

782  ^  (70  +  8)2  =  702  +  2  X  70  X  8  +  82  =  g084. 

2.  TJie  square  of  a  number  expressed  by  a  single  figure  con- 
tains no  figure  of  a  higher  order  than  tens. 

For  9  is  the  largest  number  which  can  be  expressed  by  a  single 
figure,  and  92  =  81. 

3.  The  square  of  tens  contains  no  significant  figure  of  a  lower 
order  than  hundreds,  nor  of  a  higher  order  than  thousands. 

Thus,  102  _  100,  and  902  _  gioo. 

4.  Hie  square  of  a  number  contains  twice  as  many  figures  as 
the  member,  or  twice  a»  many  less  one.     Thus, 

12=       1,  102=        100, 

92=     81,  1002=     10000, 

992  =9801.  10002  =1000000. 

Hence, 

h.  If  a  numher  be  ^rvarated  into  periods  of  two  figures  each, 
beginning  at  units'  place,  fhfi  member  of  periods  tuill  be  eqticl  to 
the  number  of  figures  in  the  square  root  of  that  number.  Thus, 
there  are  two  figures  in  the  square  root  of  43,56. 

1.  Let  it  be  required  to  extrsc*;  the  squi^re  root  of  4356. 

Let  a  represent  the 

value  of  the  first  figure  43,56(60  +  6^^0^ 

of  the  root,  and  b  that  cz2— 36  00 

of  the  second  figure.  2«  +  J=120  +  0=i26')7^ 

Since  56  cannot  be  a  2a6  +  ^2-^7  55 
part  of  the  square  of  the 

tens  (Peik.  3),  a  must  be  the  greatest  multiple  of  ten  whosf* 

Equare  is  less  than  4300 ;  this  is  found  to  be  60.     Subtracting  a" 


EVOLUTION-.  157 

that  is,  3600,  from  the  given  number,  we  find  the  remainder  to  be 
756.  This  remainder  consists  of  ttvice  the  product  of  the  tens  hy 
the  units,  and  the  square  of  the  units  (Prin.  1).  But,  since  the 
product  of  tens  by  units  cannot  be  of  a  lower  order  than  tens,  the 
last  figure,  6,  cannot  be  a  part  of  twice  the  product  of  the  tens  by 
the  units;  this  double  product  must  therefore  be  found  in  the 
part  750. 

Now,  if  we  double  the  tens  and  divide  750  by  the  result,  the 
quotient,  6,  will  be  the  units'  figure  of  the  root,  or  a  figure  greater 
than  the  units'  figure.  This  quotient  figure  cannot  be  too  small, 
for  the  part  750  is  at  least  equal  to  twice  the  product  of  the  tens 
by  the  units ;  but  it  may  be  too  large,  for  750,  besides  the  double 
product  of  the  tens  by  the  units,  may  contain  tens  arising  from 
the  square  of  the  units  (Prin.  2). 

To  ascertain  if  the  quotient,  6,  is  correct,  we  add  it  to  120  and 
multiply  the  sum  by  6.  Subtracting  the  product  from  756,  we 
find  the  remainder  to  be  0 ;  hence  66  is  the  required  square  root. 

2.  If  the  square  root  contains  more  than  two  figures,  it  may  be 
found  by  a  similar  process,  as  in  the  following  example,  where  it 
will  be  seen  that  the  partial  divisor  at  each  step  is  obtained  by 
doubling  that  part  of  the  root  already  found.  The  letters  show 
how  the  different  steps  correspond  to  those  of  the  algebraic  process 
in  Art.  266. 

a       h      c 
18,66,24(400  +  30  +  2 =432 

fl2= 160000 
2a  +  Jr=800  +  30  =  830)26624=2ff5  +  J2_|_2«c  +  2Jc  +  c3 
24900=2^^  +  ^2 
2(a-f  J)+c=:800  +  60  +  2=862)1724=2«c  +  25c  +  ^2 

1724=2ac  +  25c  +  c2 

For  the  sake  of  brevity,  the  ciphers  on  the  right  are  usually 
omitted;  thus, 

43,56(66  18,66,24(432 

36  16 

126)756  83)266 

756  249^ 

862)'l724 
1724 


loS  EVOLUTIOIT. 


RULE, 


I.  Separate  the  giveji  number  into  periods  of  ttvo  figures,  each^ 
beginning  at  the  units'  place, 

II.  Find  the  greatest  number  tchose  square  is  contained  in  the 
period  on  the  left;  this  tvill  be  the  first  figure  in  the  root.  Sub- 
tract the  square  of  this  figure  from  the  period  on  the  left,  and  to 
the  retnainder  annex  the  next  j^eriod  to  form  a  dividend. 

III.  Divide  this  dividend^  omitting  the  figure  on  the  right,  by 
double  the  part  of  the  root  already  found,  and  annex  the  quotient 
to  that  part,  and  also  to  the  divisor ;  then  multiply  the  divisor 
thus  completed  by  the  figure  of  the  root  last  obtained,  aiid  subtract 
the  product  from  the  dividend. 

IV.  If  there  are  more  periods  to  he  brought  dow7i,  continue  the 
operation  in  the  same  mariner  as  before. 

EXAMPLES. 

Find  the  square  root  of  each  of  the  following  numbers : 

1.  177241.  Ans.  ^9A. 

2.  4334724.  -  ^?i5.  2082. 

3.  14356521.  Ans.   3789. 

4.  17.338896.  Ans.   4.164. 

5.  2.5.  Ans.   1.5811+. 

270.  To  find  the  cube  root  of  a  polynomial. 
{a^\-b^ircY=a^■^^a^b  +  ^aJyi■\-b^-\-Z{a-\-bYc-{■^a■\-l)c^-\-c^. 
Let  us  now  find  the  root  a  +  b  +  c  from  its  cube. 

a^J^^a:ih^^al^J^l^J^^aJ^byc+^a^\-b)(^-\-(^ 


a-\-b 
+  c 


3a24.3aJ  +  52 


dd^b  +  '6aU^  +  b^ 
da%  +  dal^-\-¥ 


3(a  +  5)2  +  3(rt  +  %  +  (^ 


^a  +  bfc-\-'d{a-\-b)c^  +  (^ 

^a+bfc+^a+by■\-(^ 


EVOLUTION.  159 

The  first  term  of  the  root  is  obtained  by  extracting  the  cube 
root  of  c^.  Subtracting  a^  from  the  given  polynomial,  and  dividing 
the  first  term  of  the  remainder  by  Za^^  we  obtain  the  second  term 
of  the  root.  Multiplying  Za^  +  Zab  +  W-  by  l,  and  subtracting  the 
product  from  the  first  remainder,  we  obtain  the  second  remainder. 
Dividing  the  first  term  of  the  second  remainder  by  ?>o?,  we  obtain 
the  third  term  of  the  root.  Multiplying  3 (a  +  J)2 + 3(«  +  l)c  +  &■ 
by  c,  and  subtracting  the  product  from  the  second  remainder,  we 
obtain  0  for  the  remainder. 

I.  Arrange  the  given  polynomial  according  to  tlie  powers  of 
one  of  its  letters  ;  then  the  cube  root  of  the  first  term  will  be  the 
first  term  of  the  root.  Subtract  the  cube  of  the  first  term  of  the 
root  from  the  given  polynomial, 

II.  Divide  the  first  term  of  the  remainder  by  the  partial  divi- 
sor, tuhich  is  three  times  the  square  of  the  first  term  of  the  root; 
the  quotient  ivill  be  the  second  term  of  the  root. 

III.  To  the  partial  divisor  add  three  times  the  product  of  the 
first  and  second  terms  of  the  root,  also  the  square  of  the  second 
term  ;  the  result  will  be  the  first  complete  divisor. 

IV.  Multiply  the  complete  divisor  by  the  second  term  of  the 
root,  and  subtract  the  product  from  the  first  remainder. 

V.  Divide  the  first  term  of  the  second  remainder  by  the  par- 
tial divisor,  which  is  three  times  the  square  of  the  first  term  of  the 
root ;  the  quotient  will  be  the  third  term  of  the  root. 

VI.  To  three  times  the  square  of  the  sum  of  the  first  and  sec- 
ond terms  of  the  root,  add  three  times  the  product  of  the  sum  of 
the  first  and  second  terms  by  the  third,  also  the  square  of  the  third 
term  ;  the  result  will  be  the  second  complete  divisor. 

VII.  Multiply  the  second  complete  divisor  by  the  third  term  of 
the  root,  and  subtract  the  product  from  the  second  remainder. 

VIII.  If  the  required  root  contains  additional  terms, proceed 
in  like  manner  until  all  the  terms  are  found. 


160  EVOLUTION. 

Cor. — ^We  may  dispense  with  the  complete  divisor,  if,  after 
each  time  that  we  find  a  new  term  of  the  root,  we  subtract  the 
cube  of  the  sum  of  the  terms  already  found  from  the  given  poly- 
nomial. 

jexampi.es. 

Find  the  cube  root  of  each  of  the  following  expressions : 

1.  a^  +  6x^y  +  12a:^2  ^  3^.  ^;^,  ^  +  2y, 

2.  a^-\-  12a^  +  48a;  +  64.  Ans,  a;  +  4. 

3.  «»  —  9a2  +  27a  —  27.  Ans,  a  —  3. 

4.  8a3  —  SQa^b  +  54a^  —  27^8.  Ans.  2a  —  dh. 

6,  afi  +  ea^  —  40a:S  +  96a;  —  64.  Ans.  a^5  +  2a;  —  4. 

6.  a«  +  6a5  4.  150*  +  20a^  +  ISa^  +  6a  +  1- 

Ans.  a^  -\-2a  -{- 1. 

7.  a;«  —  12a*  +  54a;4  _  112:2:3  ^  iQg^  _  ^Sx  +  8. 

Ans.  a^^  —  4a;  +  2. 

8.  «6  _  3^5j  4.  6a4^2  _  7«8j3  ^  6«2^  —  3a^  +  J«. 

Ans.  a^  —  ab-{-  ^. 

9.  ^3  —  Z>3  _|.  c3  —  3  (^2^  —  a^c  —  ab^  —  ac^—  V^c  +  h(P)  —  Ubc. 

Ans,  a  —  b  -\-  c. 

10.  1  —  6a;  +  21^-2  —  56a;3  ^  iiia4  _  1743.5  ^  219a;6  —  204a;74- 
144a;8  _  64^9.  Ajis,  1  _  2a;  H-  3a;2  —  4a;3, 

11.  8a;6  +  48ca;5  _^  eOc^a;*  —  80cSa?  —  90^a;2  _^  loscSo;  —  27c«. 

^;^5.  2a;2  ^  4^^;  _  3^, 

12.  a;9  —  3a;8  +  6a;^  —  10a;«+  12a;S—  12a;4^  10:6^—  6a;2+  3a;  —  L 

Ans.  a;^  —  a:2  -j-  a;  —  1. 

13.  a;9  +  6a;8  _  64a;«  —  96a;5  _^  192^4  4.  512^3  —  7G8a;  —  512. 

Ans.   a;3  _|_  2a;2  —  4a;  —  8. 

14.  8a8  _  V2a%  +  ^Qa^bc  +  6a3^>2  _  360^2^2^  _  a%^  +  5405^2^2  _p 
9a2^>3c  _  27a^c2  +  27^c3.  Ans.  2a  —  ab  +  3/./;. 


EVOLUTION.  161 

271.  To  find  the  cube  root  of  a  mimber. 

Pkixciples. — 1.  The  cube  of  a  number  contains  three  times 
as  many  figures  as  the  number,  or  three  times  as  many  less  one  or 
two.    Thus, 

13=          1,  103=                  1000, 

33=    27,  1003=     1000000, 

93=   729,  10003=   1000000000, 

993  =  907299,  100003  = 1000000000000. 

Hence, 

%  If  a  number  be  separated  into  periods  of  three  figures  each, 
beginning  at  unit^  place,  the  number  of  periods  will  be  equal  to 
the  number  of  figures  in  the  cube  root  of  that  number.  Thus, 
there  are  two  figures  iu  the  cube  root  of  405,224. 

1.  Let  it  be  required  to  extract  the  cube  root  of  405224. 

a      b 
405,224(70  +  4=74 
a3=343  000 
Sa^=    "702x3=    7^x300=14700 
3aJ=70  x4x3  =  7x4x 30=     840 
^=  42  =       16 


62224=3«2^  +  3«J2_^j3 
62224=(3a2  +  3a&  +  ^>2)j 


da^+3ab-\-l^=  15556 

Denote  the  tens  of  the  root  by  a,  and  the  units  by  b ;  then, 
since  the  cube  of  tens  contains  no  significant  figure  of  a  lower 
order  than  thousands,  a  must  be  the  greatest  multiple  of  ten 
whose  cube  is  less  than  405000  ;  that  is,  a  must  be  70.  Subtract- 
ing the  cube  of  70  from  the  given  number,  we  find  the  remainder 
to  be  62224.  Dividing  this  remainder  by  3a%  that  is,  by  14700, 
we  obtain  4  for  the  value  of  b.  Adding  Sab,  that  is,  840,  and  b% 
that  is,  16,  to  14700,  we  find  the  complete  divisor  to  be  15556. 
Multiplying  the  complete  divisor  by  4,  and  subtracting  the  pro- 
duct from  62224,  we  find  the  remaiader  to  be  0 ;  hence,  70  +  4, 
that  is,  74,  is  the  required  cube  root 

2.  If  the  cube  root  contains  more  than  two  figures,  it  may  be 
found  by  a  similar  process,  as  in  the  following  example,  where  it 


162 


EVOLUTION. 


will  be  seen  that  the  partial  divisor  at  each  step  is  obtained  by- 
multiplying  the  square  of  that  part  of  the  root  akeady  found  by  3. 

12,812,904(200  +  30  +  4=234 
8  000  000 


2002x3=120000 
200x30x3=  18000 

4812904 
4167000 

302=       900 

645904 

138900 

645904 

2302x3=158700 

230x4x3=     2760 

42=        16 

161476 

The  work  in  the  preceding  example  may  be  abridged  as 


follows ; 


12,812,904(234 


8 

22x300=     1200 
2x3x30=       180 

4812 
4167 

3^=          9 

645904 

1389 

645904 

232x300=158700 

23x4x30=     2760 

42=        16 

161476 

RULE. 

I.  Separate  the  given  number  into  periods  of  three  figures  each, 
beginning  at  the  units'  place. 

II.  Find  the  greatest  number  whose  cube  is  contained  in  the 
period  on  the  left;  this  will  be  the  first  figure  in  the  root.  Sub- 
tract the  cube  of  this  figure  from  the  period  on  the  left,  and  to  the 
remainder  annex  the  next  period  to  form  a  dividend. 

III.  Divide  the  dividend  by  the  partial  divisor,  which  is  three 
hundred  times  the  square  of  the  imrt  of  the  root  already  found; 
the  quotient  will  be  the  second  figure  of  the  root. 


EVOLUTION^.  163 

IV.  To  the  partial  divisor  add  tJiirty  times  the  product  of  the 
first  and  second  figures  of  the  root,  also  the  square  of  the  second 
figure  ;  the  result  will  he  the  complete  divisor, 

V.  Multiply  the  complete  divisor  hy  the  second  figure  of  the 
root,  and  subtract  the  product  from  the  dividend. 

VI.  If  there  are  more  periods  to  he  drought  down,  contiime  the 
operation  in  the  same  manner  as  hefore, 

EX  A  MPIjE  S. 

Extract  the  cube  root  of  each  of  the  following  numbers: 

1.  9261.  Ans,  21. 

2.  15625.  Ans.  25. 

3.  12167.  Ans:  23. 

4.  32768.  Ans.  32. 

5.  103.823.  Ans.  4.7. 

6.  884.736.  Ans,  9.6. 

7.  12.812904.  Ans,  2.34. 

8.  8741816.  Ans.  206. 

9.  2.5.  Ans.  1.357 +. 
10.  .2.  Ans.  .5848  +  . 

272.  The  Higher  Hoots  of  Quantities.— W\\m  the 

index  of  the  required  root  contains  no  prime  factor  greater  than 
3,  the  root  may  be  found  by  methods  already  explained.  In  order 
to  show  this,  it  will  be  necessary  to  prove  the  following  principle : 

The  mn^^  root  of  a  quantity  is  equal  to  the  m^^  root  of  the  n^^ 
root  of  that  quantity. 

Let  |/V«  =  r    .     .     .     (1). 

Kaising  both  members  of  (1)  to  the  m*^  power, 
Va  =  r™    .    .    .     (2). 


164  EVOLUTION. 

Kaising  both  members  of  (2)  to  the  n^^  power, 
az=r^'''    .    .    .    (3). 

Extracting  the  mn^  root  of  both  members  of  (3), 
V^a  =  r    .    .    .     (4). 

Comparing  (1)  and  (4), 

"»"/—      "*  /♦»  /— 
Va  =  4/  ya. 


Thus,  Vl6  =  |/V16  =  a/4  =  2,  V64  =  |/V64  =  ^8=2, 

1.  Extract  the  fourth  root  of  6fl2^  +  a^  —  4^85  _  4flJ3  ^  J4. 

a^ ' 

2a2-2flJ|  _4a3d  +  6fl2/^ 


2a2_4«j_}.^  I  2«2Z>2_4«Z>3_^  j4 

a2_2aJ+62|«_j 
«^ ' 

2a— b\  —2ab-\-b^ 

We  extract  the  square  root  of  the  given  polynomial,  and  thus 
obtain  a^  —  2cf J  +  ^^ ;  we  then  extract  the  square  root  of  this 
last  expression,  and  find  the  root  to  be  a  —  J ;  this  is,  therefore, 
the  fourth  root  of  the  proposed  expression. 

2.  Extra<3t  the  fourth  root  of  81a;*  —  432a:3  +  8642;2  —  768a;  + 
256.  Ans.  ±  (3a;— 4). 


EVOLUTIOK. 


165 


3.  Extract  the  sixth  root  of  Qa^b  +  l^a^h^  +  a«  +  "^^aW  + 
Iba^b^  +  ^^  +  ^a¥,  .  Ans.  ±{a  +  b). 

4.  Extract  the  eighth  root  oia^  —  16x'^ -^  lUa:^  —  US:^ -{- 
1120a;4  —  1792a;3  +  1792a;2  —  1024^  +  256.       Ans.  ±  {x  —  2). 

273.  Hoots  of  Fractions* — TJie  n^^  root  of  a  fraction  is 
a  fraction  tohose  7iumerator  is  the  n^^  root  of  the  given  numerator, 
a7id  whose  denominator  is  the  n^  root  of  the  given  denominator. 


Thus,  ^_  =  _,  for  (-)=-, 


•^74. 


SYNOPSIS    FOR    REVIEW. 


CHAPTER  X. 


§ 


POWEB 


Degree. 

Exponent, 

Perfect. 

Imperfect. 

Sign. 

Powers  of  Products, 

Coefficient. 

Power  of  Monomial.    Rule. 

Power  of  Polynomial.    Rule. 

Power  of  Fractions. 


Root 


Index. 

Imaginary  Quantities. 
Root  of  Monomial.    Rule. 


Square  Root. 


Polynomial.    Rule.    Cor.  1,  2, 

Trinomial. 

Binomial. 


Cube  Root  op  Polynomial.    Rule.    Cor. 
Higher  Roots. 
Roots  of  Fractions. 


CHAPTEE   XL 
THEORY    OF   EXPOE'EKTS, 


a"*  X  a**  =  a^' 

> 

— -  =  a»"-",    or 
a" 

1 

275.  We  have  defined  «"»,  where  w  is  a  positive  integer,  as 
the  product  of  m  factoi-s,  each  equal  to  a,  and  we  have  shown  that 


and  that 


according  as  m  is  greater  or  less  than  n.  Hitherto  an  exponent 
has  been  regarded  as  Si^jositive  integer  j  it  is,  however,  found  con- 
venient to  use  exponents  which  are  7iot  positive  integers,  and  we 
now  proceed  to  explain  the  meaning  of  such  exponents. 

276.  As  fractional  exponents  and  negative  exponents  have 
not  yet  been  defined,  we  are  at  liberty  to  define  them  as  we  please. 
For  the  sake  of  uniformity,  we  shall  give  such  a  meaning  to  them 
as  will  make  the  relation 

true,  whatever  m  and  n  may  be. 

277.  Find  the  meaning  of  a* 

ai  X  «*  =  «!  =  «  (276); 
that  is,  a*  must  be  such  a  quantity  that  if  it  be  squared,  the  re- 


sult will  be  a ;  hence,  a*  =  Va, 


THEORY    OF    EXPONENTS.  167 

378.  Find  the  meaning  of  cfi. 

a^  xa^  X  a^  :=a^^^^^=a>=a', 
a^  =  V«- 

279.  Find  the  meaning  of  a^. 

d^  X  a*  X  a^  X  a^  =  a^; 

a^  =  \/a^. 

1 

280.  Find  the  meaning  of  a",  where  n  is  a  ^^ositive  integer. 

i  1  i  l+1+l+....tontenna 

a**  X  a"  X  a"  X   ...  to  w  factors  =  a**  "   **  =a^=a; 


a"  =  V«. 


281.  Find  the  meaning  of  a",  ivhe7'e  m  and  n  are  positive 
integers. 


tn  tn  tn  tn  ,  tn  ,  tn 


—  —  —  ,  ,.       ,  — I 1 f- ...  to  n  terms 

a"  X  a"  X  a"  X  ....  to  7i  lactors  =  ««»*»  =a^; 

m  

Hence,  the  numerator  of  a  positive  fractional  exponent  denotes 
the  power  to  which  the  quantity  is  to  be  raised,  and  the  denomi- 
nator denotes  the  root  to  be  extracted. 


282.  Find  the  meaning  of  a~\ 


w  X  a"  =  a**  *  =  a. 


3-2  — 


Dividing  both  members  of  this  equation  by  a*, 


-2        «         1 

a  2  r=  —  =  -2 

a^      a^ 


283.  Find  the  meaning  of  «"%  where  n  is  any  positive  num- 
her,  integral  or  fractional. 


168  THEORY    OF    EXPONElTrS. 

Dividing  both  members  by  a^+\ 


Hence  a~^  is  the  reciprocal  of  a*». 

284.  It  follows,  from  the  meaning  which  has  been  given  to  a 
negative  exponent,  that  —  =  a"*-^  when  m  is  less  than  n,  as  well 
as  when  m  is  greater  than  n.    For,  suppose  m  less  than  n ;  then 


285.  General  Scholium. — It  thus  appears  that  it  is  not 
absolutely  necessary  to  introduce  fractional  and  negative  expo- 
nents into  Algebra,  since  they  merely  supply  us  with  a  new  nota- 
tion for  quantities  which  we  had  already  the  means  of  represent- 
ing.    Thus, 

«■»  =  V«^,    a^  =  Va^f   «*  =  a/«^  =  a^, 

,       1       _±       1  1  -111 

a-^  =  —„,  a  «  =  —  =  -— ,    a  *  =  —  =  — 

a'  J       Va  J      a^ 

If  m  is  a  positive  integer,  the  expression  a"^  is  read  tlie  mP^ 
power  of  ttj  OT  a  m^  power.    But  if  9n  is  not  a  positive  integer, 

a"»  should  be  read  a  exponent  m.    Thus,  a^  is  read  a  exponent 
two-thirds f  not  "  a  two-thirds  power,"  for  there  is  no  such  power. 

i        i  1 

286.  To  show  that  a"  x  ^"  =  («*)**. 

Let 

11  (  i      1.Y    /  ty    ( lY 

ic  =  a"  X  ^";  then  a;«  =  [a''  x  2'"/  =  V«"/   x  W)  . 

But  Uv  x\bi  =ab  (2S0); 

x^  =  ab; 
1 
whence,  x  =  (ah)^. 


THEORY    OF    EXPONENTS.  169 

ill  11  1 

Cor.  a"  X  5"  x  C^  =  {obY  x  c"  =  {ahc^.  In  like  manner 
it  may  be  shown  that 

111  1  1 

a"  X  ^**  X  c"  X  .  .  .  .  ^"  =  (abc  ....  k)^. 

Suppose  now  that  there  are  w  of  these  quantities  a,  b,  c  , , ,  k, 
and  that  each  of  them  is  equal  to  a ;  then  the  last  equation  be- 
comes 

1        ^ 
But  {a^)^=a^  (281); 


\anj    —a 


«.    That  is, 


The  m^  power  of  the  n^  root  of  a  is  equivalent  to  the  n^  root 
of  the  mP^  power  of  a. 


287.  To  show  that  ^=(1)". 

1  1 

Let     x=^,    then    a:«= /^C  =|  (380-260) 

Jn  \bn} 


whence. 

x  = 

(D* 

288. 

To  show  that  \ar 

1 

J_ 
amn. 

Let 

(  ^\~- 
X  =  \a'"/   ;    then 

X^: 

1 

whence, 

X  = 

A. 

289. 

To  show  that  a"  = 

mp 

Let    a;  =  a" ;    then    ai^  =  0^;    and    of^p  =  a"*'; 

mp 

whence,  a;  =  a"^. 


170  THEOBY    OF    EXPONENTS. 

290,  EXAMPLBS, 

1.  Simplify   {x^  x  x^)^.  Am.  x^, 

2.  Find  the  product  of  fl*,   a  %  a  S  and  a~^. 

Ans.  cT^, 

3.  Find  the  product  of  (f  )  ,    (|)  ,    and    (t^  . 

Ans,  -^Y* 
{Ixy 

4.  Multiply  a^  +  J*  +  cT^i  by  aF^  —  a^  +  h^. 

Ans.  ah~^  4-  ah^  +  a~h^. 

5.  Multiply  ic*  —  xy^  +  x^y  —  y*   by  x  -\-  x^y^  +  y, 

Ans,  x^  +  x^y  —  xy^  —  y^, 

6.  Multiply  a^  —  flS  +  a^  —  a3  +  a^  —  a  +  a*— 1  by  a^+l 

^W5.  a*  —  1. 

7.  Multiply  a"^  —  fl^  +  1  —  a~*  +  cT^  by  a^  +  1  +  a"^. 

^?i5.  a  +  d^  —1  +  a~*  4-  a~*. 

8.  Divide  a;*  —  a:?/^  +  x^y  —  y^  by  x^  —  y^,  , 

^W5.  X  -{-  y, 

9.  Divide   a:^  +  x^a^  +  a^   by  a;**  +  x^a^  +  a* 

^Tis.  x'^  —  x^a^  +  a* 

Sn  _3n  n  _n 

10.  Divide  a^  —  a   '^    by  a«  —  a  «.     .4ws.  ««  +  1  +  a"". 

11.  Divide     a^  —  o*^'  +  ab^  -  2«iz>2  4.  jf     by    «*  —  aJ^  + 
ah  —  b^.  Ans.  a  +  aH^—  l. 


12.  Simplify 


THEOKY    OF    EXPOiTENTS. 

(^  —  ax^  +  a^x  —  x^ 

a^  —  a^x^  +  Sa^x  —  dax^  +  aJx^  —  x^ 

a  +  X 


171 


Ans. 


a^  +  Sax  +  x^ 


ifi       x^       2y^ x^ 

13.  Extract  the  square  root  of  —  +  ^ — h  ~ :; — 

Ans.^  +  x^tj^--"^. 
x^  2if 

14.  Extract  the  square  root  of  4«  —  12aH^  +  9^*  +  l^a^c^ 
—  2^i^c^  +  leA  vd^s.  2Gf^  —  dl)^  +  4A 

a 

15.  If  a*  =  1%  show  that  (t)  =  «*      ;  and  if  a=z2b,  show- 
that  ^»  =  2. 


391. 


SYNOPSIS   FOR   REVIEW. 


CHAPTER  XI. 
THEORY  OF  EXPONENTS. 


f  Bam  of  theory  (276). 

Meaning  of  a',  a  ,  a% 

Meaning  of  ««,  a». 

Meaning  of  a~^,  a~*. 

General  Scholium, 

L      L  i 

Show  that  an  x  bn  —  {aby. 

1  i_ 

Show  that  —  =  1^1. 

m  mp 


CHAPTER   XII. 
EADIOAL    QUANTITIES. 


DEFINITIONS. 

292.  A  Simple  Hadical  Quantity  is  an  expression 
of  the  form  of  a  V^,  or  ah^.  Thus,  2  a/9,  3  Vs,  l\/a,  and 
^(a  _|_  lYYc  are  simple  radical  quantities. 

293.  The  Hadical  Factor  is  the  indicated  root,  and  the 
Coefficient  of  the  radical  factor  is  the  quantity  affixed  to  the 
radical  sign.    Thus,  in  the  expression  2  V^,  the  radical  factor  is 

Vo  and  2  is  its  coefficient ;  and  in  the  expression  [(«  4-  HfYCf 

the  radical  factor  is  [(«  +  J)^]     and  c  is  its  coefficient. 

If  the  coefficient  of  a  radical  factor  is  1,  it  is  usually  omitted. 

Thus,  a/3  is  equivalent  to  1  \/3. 

294.  The  Degree  of  a  simple  radical  quantity  is  denoted 
by  the  index  of  the  radical  sign,  or  by  the  denominator  of  the 

fractional  exponent.     Thus,    h  Va  and    a  (5  +  c)^    are  of  the 

second  degree ;    h  a/«    and    a  (J  +  c)"*    are  of  the  third  degree ; 

Ja/»  and  a  ( J  +  c)^  are  of  the  fourth  degree ;  and  so  on. 

295.  Two  or  more  simple  radical  quantities  are  said  to  be 
Similar  if  their  radical  factors  are  identical.  Thus,  2  %/Z  and 
4  a/3  are  similar. 

296.  A  simple  radical  quantity  is  said  to  be  in  its  Simplest 
Form  when  the  quantity  under  the  radical  sign  is  entire  and 


EEDUCTION.  173 

does  not  contain  a  factor  which  is  a  perfect  power  corresponding 
to  the  degree  of  the  indicated  root.  Thus,  3  a/5  is  in  its  simplest 
form. 

297.  A  national  Quantity  is  one  which  may  be  ex- 
actly expressed  without  using  the  radical  sign  or  a  fractional  ex- 
ponent.   Thus,  5,  —  3,  42,  and  a  +  b  are  rational. 

Any  rational  quantity  may  be  expressed  under  the  form  of  a 

radical  quantity.     Thus,  5  =  a/25,  —  3  =  —  a/9,  4?=V^%  and 

298.  An  Irrational  Quantity  is  one  which  cannot  be 
exactly  expressed  without  using  the  radical  sign  or  a  fractional 

exponent.     Thus,   3 a/8,   2\/d,  and   5 a/3  are  irrational. 

Irrational  quantities  are  sometimes  called  surd  quantities,  or 
simply  surds. 

REDUCTION  OF  SIMPLE  RADICAL  QUANTITIES. 

299.  TJie  deduction  of  radical  quantities  consists  in 
changing  their  forms  without  altering  their  values. 

300.  To  reduce  a  rational  quantity  to  a  radical 
quantity  of  the  n^^  degree. 

3  =  a/9  =  V^  =  a/81  =  V3^; 


and  a  -{-  x=  V(«  +  xY  =  V{a  +  x)\ 

MULE. 

Raise  the  given  quantity  to  the  n^^  power  and  indicate  the  n*^ 
root  of  the  result, 

EXAMPLES, 

1.  Reduce  2a2  to  a  radical  quantity  of  the  third  degree. 

Ans,  %/W. 

2.  Reduce  a  -\-  x  to  a  radical  quantity  of  the  fifth  degree. 

Ans.  ^/(a  +  x)^. 


174  RADICAL    QUAIfTITIES. 

3.  Reduce  -  to  a  radical  quantity  of  the  sixth  degree. 

Ans.  y  —z. 

4.  Reduce   -~  6a^b  to  a  radical  quantity  of  the  third  degree. 

Ans.  V— 125^6^8. 

5.  Reduce   —  {x  -^  y)   to  a  radical  quantity  of  the  fourth  de- 
g^e®-  Ans.  -  V(x  +  y)K 

6.  Reduce   (a  —  hf  to  a  radical  quantity  of  the  w^^  degree. 

Ans.  V(a  —  by^, 

301.  To    introduce   the    coeflacient    of  the    radical 
factor  under  the  radical  sign. 

4a/2  =  vTg  X  V^  (297)  =  V32  (286) ; 
aVx  =  Vc^  X  Vx  =  V^; 

fl/=(fly)*; 

X  %/%a  —  x  —  V^  X  V^a  —  x  =  V2aa^  —  x^. 

Hence,  denoting  the  degree  of  the  radical  quantity  hy  n,  we 
have  the  following 

It  ULE, 

Multiply  tlie  quantity  under  the  radical  sign  ly  the  n^^  poicer 
of  the  coefficient  and  indicate  the  rtP*-  root  of  the  product. 

Cor. — In  a  similar  manner  any  factor  of  the  coeflBcient  may 
be  placed  under  the  radical  sign.     Thus,  3x2  a/5  =  3  V^O- 

EXAMFJLMS. 

Introduce  the  coefficient  of  the  radical  factor,  in  each  of  the 
following  expressions,  under  the  radical  sign : 

1.  6 a/5.  Ans.  a/180. 

2.  3a/3T  Ans.  V243. 


KEDUCTIOif.  176 

3.  (a  +  d)  Va+^.  Ans,  V{a  +  by. 


4. 


ay  -.  Ans.  ^/ab. 


'■^''-'yy^^T^-  Ans.V6al^. 


^   a./c   ,   a  .        ,/a  ,   a^ 

6.  -y  -  +  -.  Ans,  y  -  +  35. 

7.  5a\/bc,  Ans.  ^/b^w^c. 


8.  6x  V25^,  Ans,  V5«+2a;»-2 

9.  3a?»  {x  —  y)^.  Ans.  {2W  —  272:62^)* 


1 

f2 


302.  To  remove  a  factor  from  under  the  radical 
sign  to  the  coeflacient. 

The  reduction  is  performed  by  reversing  the  process  of 
Art.  301, 

Thus,      2a/8  =  2\/4xV2  =  2x2a/2,     and     a  VWc  = 

ab  ^/c. 

Henee,  denoting  the  degree  of  the  radical  quantity  by  n,  we 
have  the  following 

It  ULE. 

Divide  the  quantity  under  the  radical  sign  by  the  factor  to  be 
removed;  and  to  the  indicated  n^  root  of  the  quotient  prefix,  as  a 
coefficient,  the  product  of  the  given  coefficient  and  the  n^  root  of 
tliefa^ctor  to  be  removed. 

JEXAMPTjES. 

1.  Keduce  \/20  to  such  a  form  that  the  factor  4  shall  not 
occur  under  the  radical  sign.  Ans.  2  Vs. 


176  RADICAL    QUANTITIES. 

2.  Reduce  V24  to  such  a  form  that  the  factor  8  shall  not 
occur  under  the  radical  sign.  4,^,,^  2  V^. 

3.  Reduce  3  'v/TS  to  such  a  form  that  the  factor  25  shall  not 
occur  under  the  radical  sign.  j^^^g  15  a/3^ 

4.  Reduce  (a  —  h)  V(a  +  h^c  to  such  a  form  that  the  factor 
{a  +  by  shall  not  occur  under  the  radical  sign. 

Ans.  (a2  —  ^)  a/c. 

5.  Reduce  a  (5  +  c)  V^  to  such  a  form  that  the  factor  d" 
shall  not  occur  under  the  radical  sign. 

303.  To  reduce  the  indicated  root  of  a  fraction  to 
an  equivalent  expression  in  which  the  quantity  under 
the  radical  sign  shall  be  entire. 

|/^  =  j/lTe  =  l/|  X  V6  (286)  =  I  a/6 ; 

l/n     l/nTl     1/3     '/I     :     »/r     ,/-     I3,- 
'Ai     "./T^TS^^     \fa^^     "/I      T"     "/T     .__ 


n-l 


Hence,  denoting  the  degree  of  the  radical  quantity  hy  n,  we 
have  the  folio wiug 

RULES. 

I.  If  the  fraction  under  the  radical  sign  has  a  denominator 
which  is  a  perfect  n^^  poiver,  prefix  to  the  indicated  n^^  root  of  the 
numerator  the  reciprocal  of  the  n^^  root  of  the  denominator. 

n.  If  the  fraction  under  the  radical  sign  has  a  denominator 
which  is  not  a  perfect  n^^  power,  multiply  or  divide  loth  of  its 


REDUCTIOiT.  177 

terms  ly  such  a  quantity  as  will  reduce  it  to  one  wliose  denomina- 
tor is  a  perfect  n^^  power ;  then  substitute  this  fraction  for  the 
given  one  and  jJroceed  as  directed  in  I. 

Cor.— If  the  given  radical  quantity  has  a  coefficient,  the  re- 
sult obtained  by  the  rule  must  be  multiplied  by  it,  in  order  to 
obtain  an  expression  equivalent  to  the  given  one.    Thus, 

JEXAMPLES. 

Reduce  each  of  the  following  expressions  to  another  in  which 
the  quantity  under  the  radical  sign  shall  be  entire : 

1.  V^-  A7is.^V6. 

2.  ||/f-  Ans.lVm 

3.  2|/|-  Ans,  a/3. 

4.  IVIf.  Ans.\VT^. 

0  0 

5.  -^V^-  Ans.,^VlSu. 


6.     m{a  -\-  x)\  ^^~~    •  Ans,  m  V(«  —  a;)  («  +  x)''-\ 

a  "I"  3/ 

304.  To   reduce   a   simple   radical  quantity  to   its 
simplest  form. 

1.  Reduce  3  VS  to  its  simplest  form. 

The  largest  perfect  square  which  is  a  factor  of  8  is  4.    Remov- 
ing this  factor  from  the  radical  sign  to  the  coefficient,  we  have 

dVS  =  6V2  (303). 


178  KADICAL    QUAKTITIES. 

2.  Reduce  5  \/4:Sa^x^  to  its  simplest  form. 

The  largest  perfect  cube  which  is  a  factor  of  ASa^x*-  is  Sa^ofi, 
Bemoving  this  factor  from  the  radical  sign  to  the  coefficient,  we 
have  

3    /g 

3.  Reduce  4  y  ^  to  its  simplest  form. 

4  V^  =  4i/|  =  I  V2i  (303)  =  I  X  2  V3  (302)  =  |  Vs. 

Hence,  denoting  the  degree  of  the  radical  quantity  by  7i,  we 
have  the  following 

MULBS. 

I.  If  the  quantity  under  the  radical  sign  is  e7itire,  resolve  tt 
into  two  factors,  one  of  which  is  the  greatest  n^  poiver  contained 
in  that  quantity ;  then  remove  this  factor  from  the  radical  sign 
to  the  coefficient. 

n.  If  the  given  radical  quantity  contains  the  indicated  root  of 
a  fraction,  reduce  it  to  such  a  form  that  the  quantity  under  the 
radical  sign  shall  he  entire,  and  proceed  with  the  result  as  directed 
in  I. 

EXAMrms. 

Reduce  each  of  the  following  radical  quantities  to  its  simplest 
form: 

Ans.  5a  Vah. 

Ans.  da^cVdbx, 

Ans.  Sa^^c^V^. 

Ans.  3a  \^. 

Ans.  Qx  \/a  -f  ba?. 

Ans.  35al}\/c. 


1. 

V25a^. 

2. 

V'^Wl^ch;- 

3. 

4. 

Vl92a^^cl 
VlOSa^. 

5. 

6  Vax^  4-  ^^. 

e. 

7V625a*Z>V. 

REDUCTIOJS^.  179 


7.      3  Va^+"^.  Ans,  3a  Va^'b, 


8.  V(a  +  x)H**.  Ans,  (a  +  x)  V6". 

9.  ||/|.  Ans.^Vn, 


10.       6 


|/|-  ^W5.  2  Vis. 


11.       t1/t*  -47i5.  TiV^d, 

b  ^   d  Id 


12.  2 1/,/^    ,'  ^^&  -^-  VC^  +  ^)  «. 

13.  (— )  .  Ans,  —  (a'^l^T^y^)^, 
\xy  /  XV 


14 


xyf  xy 


^^^      ^/5(^^^^).  ^;^..  ^3V5TrT^)^. 


16.  (a^b)\/^—4'  Ans.  V^^::^, 

a  -\-  0 

17.  (a  —  b)  \/ ' ; Ans,  ^/cim  +  n). 

305.  To  reduce  a  radical  quantity  of  the  form  of 
V^  to  another  of  a  lower  degree. 

V9  =  |/V^(272)  =V3; 

V8  =  /Vi='V/2; 

mrij —         *"  /n  / —         m  /— 


180  RADICAL    QUANTITIES. 

Hence,  denoting  the  factors  of  the  index  by  m  and  n,  we  have 
the  following 

RULE. 

Extract  the  n^^  root  of  the  quantity  under  the  radical  sign, 
and  indicate  the  rrif^  root  of  the  result, 

EXAMJPLES. 

1.  Reduce   V^a^  to  a  radical  quantity  of  the  third  degree. 

Ans.  a/^«- 


2.  Eeduce  v64a^  to  a  radical  quantity  of  the  second  de- 
gree, j^ns.  V^i' 

3.  Reduce  \/lM¥c^  to  a  radical  quantity  of  the  second  de- 
gree, jins.  ^/%aM, 

4.  Reduce   V^oa^^c*  to  a  radical  quantity  of  the  third  degree. 

Ans.  ^/hab(^. 
6.  Reduce   \f^J^(^  to  a  radical  quantity  of  the  third  degree. 

Ans,  VoW. 

6.  Reduce   Va^^  to  a  radical  quantity  of  the  fifth  degree. 

Ans.  V«^c3. 

306.  To  reduce  a  simple  radical  quantity  to  another 
of  a  higher  or  lower  degree. 

Vff  =  a*  =  a^  =  a^  =  «^  =  a'*  =  v^a»; 

'^^«  =  a^=a^  =  c.*  =  a*  =  a*  =  V«. 

nTJL,B. 

Express  the  given  radical  quantity  hy  means  of  a  fractional 
exponent;  then  suMitute  for  this  exponent  any  equivalent  frac- 
tion having  a  denominator  greater  or  less  than  that  of  the  given 
fractional  exponent. 


KEDUCTION.  181 

Cor.  1.— If  equal  factors  be  introduced  into  the  index  of  the 
root  and  the  exponent  of  the  quantity  under  the  radical  sign, 
the  result  will  be  equal  to  the  given  radical  quantity.    Thus, 

CoE.  2. — Conversely,  if  equal  factors  be  canceled  in  the  index 
of  the  root  and  the  exponent  of  the  quantity  under  the  radical 
sign,  the  result  will  be  equal  to  the  given  radical  quantity.    Thus, 

EXAMFZJES. 

1.  Reduce   V«  to  a  radical  quantity  of  the  12th  degree. 

Ans.  V«®. 

2.  Reduce  *Va  to  a  radical  quantity  of  the  mn^^  degree. 

Ans.   va". 

3.  Reduce    Va  —  b  to  a  radical  quantity  of  the  20th  degree. 

Ans.  ^\/(a  —  by. 


4.  Reduce    a/(«  +  by  to  a  radical  quantity  of  the  10th  degree. 

Ans.  ^^/(a  +  by. 


5.  Reduce    V{a  —  by^  to  a  radical  quantity  of  the  3d  degree. 

Ans.  V(a  -  by. 

307.  To  reduce  simple  radical  quantities  having 
unequal  indices  to  equivalent  ones  having  equal  in- 
dices. 

1.  Reduce  Va  and  \/a  to  equivalent  expressions  having 
equal  indices. 

Va  =  a^  =  J=  V^; 

and  Va  =  a^  =  a^=  V^. 

2.  Reduce  v^a  and  ^/a  to  equivalent  expressions  having 
equal  indices. 


182  KADICAL    QUANTITIES. 

%  =  a-  =  a"-  =  T^; 

JRULE. 

Express  the  indicated  roots  by  means  of  fractional  exponents, 
and  reduce  the  expressions  thus  obtained  to  equivalent  ones,  in 
which  the  fractional  exponents  shall  have  equal  denominators, 

EXAMPLES, 

1.  Reduce  v^  and  3  V3  to  equivalent  expressions  having 
equal  indices.  j^^.  »^  and  3  '^/m. 

2.  Reduce  a/2,  VS?  V^,  and  %/b  to  equivalent  expressions 
having  equal  indices.  j^^s,  ^2*,    v^,    ''V^%    ^^/^. 

3.  Reduce  \/2,  V3,  V^,  and  V^,  to  equivalent  expres- 
sions having  equal  indices. 

Ans,   v'ioge,    v^729,    ''V/626,    '^v^sS. 

4.  Reduce  3*  2*^,  and  5*  to  equivalent  expressions  having 
equal  indices.  ^^^_  3^   ^-h^  5A 

i  1 

5.  Reduce  a»»  and  J"*  to  equivalent  expressions  having  equal 

6.  Reduce  Vox,  Vxy,  and  Vc^  to  equivalent  expressions 
having  equal  indices. 

Ans.  *v^^^^^    "v"^^;    ^"a/^^^. 

7.  Reduce  V^,  V^,  V?,  V^,  and  T  6^  to  equivalent 
expressions  having  equal  indices. 

Ans.    a/«,    ^/h,    ^~c,    Vd,    \/e. 


COMBINATIONS.  183 

COMBINATIONS  OF  RADICAL  QUANTITIES. 

308.  To  find  the  STim  of  two  or  more  simple  radi- 
cal quantities. 

1.  Find  the  sum  of  6  V2  and  8  V^. 

6  v^  +  8  VS  =  (6  +  8)  a/2  =  14  V^. 

2.  Find  the  sum  of  2  V^i  and  3  Vl92. 

2V24    =2V8^3    =    4V3, 
and  3  VTd2  =  3  V64  x  3  =  12  V^  ; 

2V24    +3Vl9^        =:16V3. 

3.  Find  the  sum  of   V^^,    V^^,    and    a/«^^. 
V^   =    a;  V^, 

V^^  =  2a;  V^, 
and  'v/^=  aVii^; 
.-.     V^+  V4^+ V^=^A/^+2a;  V54-aV^=(3a;  +  a)\/i. 

4.  Find  the  sum  of  2  Vi08  and  5  V24. 


2Vl08  =  2V27  X  4=    6V4, 

and  5V24    =5V8ir3    =10V3; 

2V1O8  +  5V24         =    6V4  +  IOV3. 

In  this  example  the  radical  quantities  cannot  be  made  similar; 
hence  the  addition  can  only  be  indicated. 

5.  Find  the  sum  of  2  V36  and  3  a/6. 

3  a/6  =  3  V36  (306,  Cor.  1). 

.-.     2  V36  +  3  a/6  =  2  a/36  +  3  a/36  =  5  V36  =  6  V^  (305). 


184  BADICAL    QUANTITIES. 


RULES. 

I.  If  the  given  radical  quantities  are  similar,  prefix  the  sum 
of  the  coefficients  to  the  common  radical  factor. 

II.  If  the  given  radical  quantities  are  of  the  same  degree,  but 
not  similar,  reduce  them,  if  possible,  to  equivalent  similar  ones  by 
the  rule  of  Art.  304,  and  proceed  with  the  results  as  directed 
in  I.    If  they  cannot  be  so  reduced,  indicate  their  sum. 

in.  If  the  given  radical  quantities  are  of  different  degrees, 
reduce  them  to  equivalent  ones  of  the  same  degree,  and  proceed 
with  the  results  as  directed  i7i  IL 


EXAMPZE8. 

1.  Find  the  sum  of  7  VlO  and  2  \/90.  Ans,  13  a/To. 

2.  Find  the  sum  of  VsOO  and  V256.  Ans.  9  Vi 

3.  Find  the  sum  of  4  -^^500  and  3  v'lOS.  Ans.  29  Vi. 

4.  Find  the  sum  of  |/ ^ >   r  5»  ^^^  V  Tq'  Ans.  \/2. 

Z  V  lo 

6.  Find  the  sum  of  kV  qj  7  r  -g"?   8,nd    gV  05* 

6.  Find  the  sum  of  Va^x  and  V^  Ans.  (a  -\-  b)  ^/x. 

7.  Find  the  sum  of  Vu,  2  \/72,  and  a  V^. 

Ans.  2  ( Ve  +  6 a/2)  +  ax  a/6. 

3    /o>v„^^  3 


8.  Find  the  sum  of  y  -^r-  and  y  —r . 


3^4-1 3 


Ans,  --^  A/4a%. 


COMBINATIOIfS.  185 

9.  Find  the  sum  of      V(l  +  a)-\       Va^  (1  +  a)~\      and 
a  V(l  +  «)(!  +  ay.  A71S.  {a^ -{-  a  +  1)  Vl  -\- a. 

10.  Find  the  sum  of  3  VlQa^^  and  5  V^ab?. 

Ajis,  (3c  -\-  5c2)  ^tabc. 

11.  Find  the  sum  of  ^/%a:j?—^ax-\-%a  and  ^/%a7?-\-^ax-\-%a, 

Ans,  2xV^ci- 


12.  Find  the  sum  of   V^a'^^^b%    Vl^a^'~^b%    V2a4"'+»,   and 
^^^'  Ans.  (Sa^  +  ^  +  a-»+3  +  c)  ^2^, 

13.  Find  the  sum  of  a  V^  and  c  Vb^- 

Ans.  ahV^+cI^Vb. 

14.  Find  the  sum  of  «  (l  +  ^)    and  b(l  +  ^)  . 

'  Ans.[(a^  +  b^)T 

309.  To  find  the  diflference  of  two   simple  radical 
quantities. 

1.  Subtract  6  V2  from  8  V2. 

SV2  —  6V2  =  {8-6)V2  =  2V2. 

2.  Subtract  2  V^i  from  3  Vl92. 

3  a/T92  =  3  V64  X  3  =  12  V3, 

and  2V24    =2V8~>r3    =    4V3; 

3V192  — 2V24         =    8V3. 

3.  Subtract  V^  from  V^- 

a/4^  =  2xVx, 

and  V?  =   xVx; 


186  RADICAL    QUANTITIES. 

4.  Subtract  2  \/l"08  from  5  V^i. 

sV^i   =5V8^3    =10V3, 
and  2Vi08  =  2V27~xl=    6V4; 

5V24  — 2V108      =ioV3  — eVi. 

In  this  example  the  radical  quantities  cannot  be  made  similar; 
hence  the  subtraction  can  only  be  indicated. 

6.  Subtract  2  V36  from  3  \/6. 
3^/6  =  3^/30(306,00^1); 
...    3a/6  — 2  V36  =  3V36-2V36  =  V36  =  >v/6  (305). 

MULES. 

I.  If  the  given  radical  quantities  are  similar,  subtract  the  co- 
efficient  of  the  radical  factor  ^71  the  subtrahend  from  that  of  the 
radical  factor  in  the  minuend,  and  prefix  the  remainder  to  the 
common  radical  factor, 

II.  If  the  given  radical  quantities  are  of  the  same  degree,  but 
not  similar,  reduce  them,  if  possible,  to  equivalent  similar  ones  by 
the  rule  of  Art,  304,  and  proceed  with  the  results  as  directed  in 
I,    If  they  cannot  be  so  reduced,  ijidicate  the  subtraction. 

in.  If  the  given  radical  quantities  are  of  different  degrees, 
reduce  them  to  equivalent  ones  of  the  same  degree,  and  proceed  with 
the  results  as  directed  in  11. 

EXAMPLES, 

1.  Subtract  ^/ba  from  V^a.  Ans.  2  ^/Ea. 

2.  Subtract  ^/U  from  ^^^192.  Ans.  2  V3. 

3.  Subtract  ^V^W  from  3a  Vb,  Ans.  2a  Vb. 

4.  Subtract  3  V^  from  6  V^.  Ans.  3  Va. 


COMBII^^ATIONS.  187 


6.  Subtract  (a  -  x)  |/^-±^  from  {a  -  x)  Va^  -  x\ 


a  —  x 


Ans.  (a  —  x  —  l)  ya^  —  x^. 


6.  Subtract  i/     ^      .,  ,    .   ,p    from  y  -^ — ^.    ,   .o  - 

7.  Subtract  V32«  from  2  V40«.       Jw5.  4  VSa  —  2  V2a. 

8.  Subtract  2  V54  fi-om  6  V320.  Ans.  24  Vs  -  6  V^- 

310.  To  find  the  product  of  two  or  more  simple 
radical  quantities. 

1.  Multiply  6  \/54  by  3  ^2. 

6  \/54  X  3  V2  =  6  X  3  V54  x  \/2  =  18  V54  x  2  (286)  = 
18  VT08  =  108  Vs. 

2.  Multiply  3  ^/U  by  2  V3«. 

3  V^  x  2  V3«  =  3x2  ^/^  x  V3^  =  6  V(2^  X  V(3^ 
(306,  Cor.  1)  =  6  V(2«)M3«)^  =  6  V72^^. 

RULES. 

I.  i/*  /^e  ^iVe»  radical  quantities  are  of  the  same  degree,  find 
the  product  of  the  radical  factors  ty  the  principle  of  Art.  286, 
and  to  the  result  prefix  the  product  of  their  coefficients.  Express 
the  final  result  in  its  simplest  form. 

II.  If  the  given  radical  quantities  are  of  different  degrees,  re- 
duce them  to  equivalent  ones  of  the  same  degree,  and  proceed  with 
the  results  as  directed  in  I. 

Cor.  1. — If  the  roots  are  indicated  by  fractional  exponents, 
the  product  may  be  found  by  the  principles  of  Ari  69. 

Cor.  2. — The  product  of  two  or  more  simple  radical  quantities 
can  always  be  reduced  to  a  simple  radical  quantity. 


188  RADICAL    QUANTITIES. 

1.  Multiply  i  a/6  by  -ft  V9.  Ans.  -^  VEi  =  ^  \/6. 

2.  Multiply  4y  I  by  3|/|.  Ans.  4^/15. 

3.  Multiply  3  -v^  by  4  V3.  Ans.  12  Vi32. 

4.  Multiply  a/24^  by  Vl2x.  Ans.  12a  x^2. 

5.  Multiply  d  \/ax  by  c  Vxy.  Ans.  be  Va^y^ 

6.  Multiply  3  V^  by  4  ^a.  Ans.  12  '\/^. 

7.  Multiply  (rt  +  i)^  by  (a  +  ^>)i.  Ans.  a -\- b. 

8.  Find  the  product  of  V^bc~\    V«,  VS"^*^^,  and   V«~i. 

9.  Multiply  (a  +  5)*  by  {a  —  V)^.  Ans.  (a^  —  b^)^. 

10.  Multiply  aVi  by  b^^/y.  Ans.  aZ>  V^aj^'y^ 

311.  To  find  the  product  of  polynomial  radical 
quantities. 

The  product  of  two  polynomial  radical  quantities  is  found  by 
combining  the  rules  of  Art.  310  with  that  of  Art.  70.  If  frac- 
tional exponents  are  used  to  indicate  the  roots,  the  rule  of  Art.  70 
is  sufficient. 


EXAMPLES. 

by  2  —  Vs. 

3+     \/5 

2—     V5 

6  +  2\/5 

-  3  V5  -  5 

Product,  1  —     a/5. 


COMBINATIOKS.  189 

2.  Multiply  x  +  2Vy  +  sVz  hy  x  —  2Vy-\-  Z\fi, 

ic  +  2  V^  +  3  v^ 
X—  2  v^  +  3  V^ 

—2x  Vy  —^y  —  6  Vy  x  \/z 

■i-dxVz  +  QVy  xVz  +  ^V^ 
Product,     aj2_4y  j^exVz-\-9  V?. 

3.  Multiply  J  +  a^b^  +  ah  +  ^'^  by  d^  —  bi 

a^  +  a^b^  4-  ah  +  b^ 


a    +  aJ^>i"  +  ah  +  a^Z^^ 
_  ahi  —  ah  —  ah^  —  &2 
Product,  a  —  b"^. 

4.  Multiply  V8  +  \/3  by  a/8  —  V3.  ^W5.  6. 

5.  Multiply  V^+  V^  by  Va  —  2^/l^, 

Ans.  a^  +  Va^*  —  2  V^  —  2  V^ 

6.  Multiply  Va  +  Vb  -\-  x  by  V^  —  ^b  +  a;. 

^Tis.  a  —  b  —  x, 

7.  Multiply  a^  -  2a^^  +  ^a^^'"*  —  8a5  4-  IGa^Z**  —  32Z>'^  by 
fli  -f  2J^.  ^W5.  «8  —  642>2. 

8.  Multiply  a^  +  a^b^  +  Z>^  by  a*"  —  Z**  ^W5.  a  —  J. 

9.  Multiply  x^y  +  y^  by  a;*  —  y  ».  ^W5.  x^y  —  y^. 

312.  To   find  the   quotient   of  two   simple  radical 
quantities. 

1.  Divide  6  ^54  by  3  \/2. 
3v^         V^ 


190  EADICAL    QUANTITIES. 

2.  Divide  8  A/2a  by  2  Va. 

RULES. 

I.  If  the  given  radical  quantities  are  of  the  same  degree, 
divide  the  radical  factor  in  the  dividend  hy  that  in  the  divisor, 
hy  the  principle  of  Art.  387,  and  to  the  result  prefix  the 
quotie)it  obtained  by  dividing  the  coefficieiit  in  the  dividend  by 
that  in  the  divisor.  Express  the  final  result  in  its  simplest 
form. 

II.  If  the  given  radical  quantities  are  of  different  degrees, 
reduce  them  to  equivalent  ones  of  the  same  degree,  and  proceed 
with  the  results  as  directed  in  I. 

Cor.  1. — If  the  roots  are  indicated  by  fractional  exponents, 
the  division  may  be  performed  by  the  principles  of  Art  84. 

Cob.  2. — The  quotient  of  two  simple  radical  quantities  can 
always  be  reduced  to  a  simple  radical  quantity. 

1.  Divide  8  \/l08  by  Vq.  Ans.  24  V^. 

2.  Divide  V512  by  4  V2.  Ans.  VI. 

3.  Divide  12  V54  by  3  V2.  Ans.  12. 

4.  Divide  4  Vi2  by  2  V3.  Ans.  |  Vl6  x  3s. 

5.  Divide  a  by  Va.  Ans.  Va. 

6.  Divide  Va  by  V^.  Ans.  yVab'^K 

be 


7.  Divide  2aWby  4vWc^.  Ans.  ^.^/V'cd^. 

2d 


COMBIlfATIONS. 

1£ 

8.  Diyide  |/|  by  j/|. 

a 

9.  Divide  cfi  by  a*. 

Ans,  d^. 

10.  Divide  a»  by  «g. 

mq-np 

Ans,  a  nq  . 

313.  To  find  the  quotient  of  two  polynomial  radical 
quantities. 

The  quotient  of  two  polynomial  radical  quantities  is  found  by 
combining  the  rules  of  Art.  313  with  that  of  Art.  86.  If  frac- 
tional exponents  are  used  to  indicate  the  roots,  the  rule  of  Art. 
86  is  sufficient. 

EX  AMPLE  8. 

1.  Divide  V^  —  a/«^  —  Va  +  V«  by  \/a  —  1. 
\/a^  —  "s/a  —  \/a^  +  \/a  \  \/a  —  1 

3.  Divide  a^  +  ^ah^  -  4.Jb'^  -  Sh^  by  J—  4.A 

gi  _  4at^,i J+2b^ 

2a^b^-  Sb^ 
2ah^-  Sb^ 

3.  Divide  a^  +  aVb  —  6b  hy  a —  2Vb.      Ans.  a  +  SVb. 

4.  Divide  a  —  41  V«  -  120  by  V^  +  4  V«  +  5. 

Ans.  V^-4V^+llV«-24. 

5.  Divide  x^  —  a^x^  —  4«a:  ^+  6a^x  —  2a^xi  by  x^  —  4^:?;^  + 
2a*.  ^ws.  ic  — «'^a:». 


192  RADICAL    QUANTITIES. 

6.  Divide  x^  —  y*  by  2;^  —  y*.  Ans.  x^  +  yi. 

7.  Divide  x^  —  %x*y*  +  1/^   hy  x*  —  y'^.        Ans,  a;*  —  y^. 

INVOLUTION  OF  RADICAL  QUANTITIES. 

314.  To  find  any  power  of  the  indicated  n^  root  of 
a  Quantity. 

( Vs)'  =  V3  X  V3  =  V32  (310) ; 

(V3)'=V32x  V3=  V38  =  3; 

(Va)"*  =  V^  (286,  Cor.). 

Hence,  denoting  the  index  of  the  given  indicated  root  by  n, 
we  have  the  foUowing 

RULE. 

Raise  the  quantity  under  the  radical  sign  to  the  required 
power,  and  indicate  the  n^  root  of  the  result. 

Cor.  1. — If  the  index  of  the  given  indicated  root  is  equal  to 
the  exponent  of  the  power  to  which  that  root  is  to  be  raised,  the 
required  power  may  be  obtained  by  simply  removing  the  radical 

sign.    Thus,  (V«)  =  a,  ( V«)  =  «j  and  (Va)   =  a. 

Cor.  2. — If  the  index  of  the  given  indicated  root  and  the  ex- 
ponent of  the  required  power  contain  a  common  factor,  the  result 
obtained  by  the  rule  may  be  reduced  to  a  radical  quantity  of  a 

lower  degi-ee.    Thus,  {V~af  =  V^  =  Va^  (306,  Cor.  2). 

CoR.  3. — If  the  root  is  indicated  by  a  fractional  exponent,  the 
ruleof  Art.  258issuflacient.     Thus,  [a^)  =a^,   (a")   =a\ 

315.  To  find  any  power  of  a  simple  radical  quan- 
tity. 

(5  V3)^  =  6V3x5V3  =  5x5xV3xV3  =  b'^V^', 
{aVby  =  a^Vb^. 


INVOLUTION.  193 

Hence,  denoting  the  exponent  of  the  required  power  by  m,  we 
have  the  following 

nULE, 

Raise  the  given  radical  factor  to  the  m^^  power  (314),  and  to 
the  result  prefix  the  m^  power  of  the  given  coefficient. 

CoR. — If  the  root  in  the  given  expression  is  indicated  by  a 
fractional  exponent,  the  rule  of  Art.  2>5S  is  sufficient    Thus, 

(2a*)  =  23a^  =  8a* 

EXAMPLES, 

1.  Find  the  square  of  5  V«-  -^ns.  25  \/a\ 

2.  Find  the  third  power  of  ha  %/x.  Ans,  125a%. 

3.  Find  the  square  of  a^  V^-  ^ns,  a*  V36. 

4.  Find  the  3d  power  of  |  Vd,  Ans,  |  a/3. 

5.  Find  the  4th  power  of  —  \/a^,  Ans.  a^  \/~o^. 

6.  Find  the  75th  power  of  x  ^Vy-  Ans.  x^  ^/y\ 

7.  Find  the  square  of  x^y.  Ans.  x^\/~y. 

8.  Find  the  n^^  power  of  x^'s/y,  Ans.  uf^^/y. 

9.  Find  the  4th  power  of  ^a  .  Ans.  -^cb^- 

10.  Find  the  6th  power  of  a  (Z>  +  c)^.    Ans.  a^  (^^+2dc4-c2). 

316.  To   find  any  power  of  a  polynomial  radical 
quantity. 

Any  power  of  a  polynomial  radical  quantity  is  found  by  com 
bining  the  rule  of  Art.  315  with  that  of  Art.  359.    If  fractional 
exponents  are  used  to  indicate  the  roots,  the  rule  of  Art.  359  is 
sufficient. 


194  RADICAL    QUANTITIES. 

EXAMPLES. 

1.  Find  the  square  of  Vs  +  a  a/2. 

Ans.  3  +  2a  Ve  +  2a^ 

2.  Find  the  third  power  of  3  +  V5.  Ans.  72  +  32  \/5. 

3.  Find  the  square  of  a*  +  b^-  ^ns.  a  +  Za^b^  +  b^. 

4  Find  the  4th  power  of  a*  —  b^. 

Ans.  a2  _  4:a^l>i  4-  6aJ  —  4ai7>^  +  P, 

EVOLUTION  OF  RADICAL  QUANTITIES. 

317.  To  find  any  root   of  the   indicated  root  of  a 
quantity. 

v¥  =  V32  (272)  =  V3  (306,  Cob.  2) ; 


m  /n  /-        mnr- 


Hence,  denoting  the  index  of  the  given  indicated  root  hy  w, 
and  that  of  the  required  root  by  m,  we  have  the  following 

L  If  the  quantity  tinder  the  given  radical  sign  is  a  perfect 
m^  power,  extract  the  m^^  root  of  it,  indicate  the  n^^  root  of  the  re- 
suit,  and,  if  possible,  reduce  to  a  lower  degree. 

IT.  If  the  qicantity  under  the  given  radical  sign  is  not  a  per- 
fect m^^  power,  indicate  the  mn^^  root  of  it,  and,  if  possible,  re- 
duce the  result  to  a  lower  degree. 

Cor. — If  the  given  indicated  root  is  expressed  by  a  fractional 
exponent,  the  rule  of  Art.  265  is  sufficient.    Thus, 


l/J = «^ 


EVOLUTION.  195 

318.  To  find  any  root  of  a  simple  radical  quantity. 


|/ 25  V32  =  5 1/ V32  =  5  V32  =  5  V3 ; 


1/5V9  =  \/yVZb  X  9  (301)  =  ''v/125  X  9  =  '^^1125 ; 


yl)Va  =  y'^^alt  =  *a^«Z>". 


Hence,  denoting  the  index  of  the  required  root  by  m,  we  have 
the  following 

I.  If  the  given  coefficient  is  a  perfect  wP^  power,  prefix  the  mP^ 
root  of  it  to  the  m^  root  of  the  given  radical  factor. 

IT.  If  the  given  coefficient  is  not  a  perfect  nnP^  power,  introduce 
it  under  the  given  radical  sign,  and  find  the  m^  root  of  the  result 
(317). 

Cofi. — If  the  radical  factor  is  expressed  by  means  of  a  frac- 
tional exponent,  the  rule  of  Art.  265  is  suJGficient.    Thus, 


|/5x  9^  =  5*  X  9* 


EXAMPLES. 

1.  Find  the  square  root  of  9  V3.  Ans.  3  V3. 

2.  Find  the  square  root  of  3  V^.  Am^^  Vl35. 

3.  Find  the  cube  root  of  |y  |  Ans,  ^V^- 

4^/4  1 3  / — 

4.  Find  the  fourth  root  of  ^  V  g.  ^ns.  ^  Vl^- 

5.  Find  the  sixth  root  of  a^  V?-  ^^^-  «^  ^^' 

6.  Find  the  fourth  root  of  a^s  V^-  ^^«-  «*  V^. 


196  RADICAL    QUANTITIES. 

7.  Find  the  fifteenth  root  of  VJaTW^-     Ans.  V(«  +  b)^ 

8.  Find  the  7i*^  root  of  c^^/a\  A7is.  a  Va. 

319.  To  find  the  square  root  or  the  cube  root  of  a 

polynomial  radical  quantity. 

The  square  root  or  the  cube  root  of  a  polynomial  radical  quan- 
tity is  found  by  combining  the  rules  of  Art.  318  with  those  of 
Articles  266  and  370. 

If  the  roots  in  the  given  expression  are  indicated  by  fractional 
exponents,  the  rules  of  Articles  266  and  270  are  sufl&cient. 

1.  Find  the  square  root  of  Vx  +  2  \/xyi  +  Vy. 
Vx-\-2  \fxy  +  v^  I  V^+  Vy 


2Va;-i-  Vy  |   ^Vxy^-y/y 
2\/xy  -\-  Vy 

2.  Find  the  square  root  of  x^  —  2x^y^  -f  ?/* 

^  __  '■tx^y^  -f  y^  \x^  —  y^ 


2a;^  —  y^    —  2icy  +  y^ 
—  2x^y'^  +  y^ 

3.  Fmd  the  square  root  of  4  Vc^  -f  12  Vab  +  9  %/¥, 

Ans.  2  V«  +  3  Vb. 

4.  Find  the  cube  root  of  «  +  3  VM  +  3  V«^  +  b- 

Ans.  %/a  +  \f'b. 


REDUCTIOIJ".  197 

5.  Find  the  square  root  of  a -{-  2a^b^  +  ^  +  2aM  +  'Zbh^ 
-\-  c.  Ans.  «i  -I-  Z>i  +  c^. 

6.  Find  the  cube  root  of  8a  +  dQah^  +  54a^<5>*  +  27h, 

Ans.  2a^  +  3^^ 

REDUCTION  OF  FRACTIONS  HAVING  SURD  DENOMINATORS  TO 
EQUIVALENT  ONES  HAVING  RATIONAL  DENOMINATORS. 

330.  A  Simple  Surd  is  a  surd  of  the  form  a\/b  or 
ab^.     Thus,  2  V3  is  a  simple  surd. 

321.  A  Polynomial  Surd  is  a  surd  having  two  or 
more  terms.     Thus,  2  a/3  +  3  \/5  —  6  \/7  is  a  polynomial  surd. 

322.  To  reduce  a  fraction  whose  denominator  is  a 
simple  surd  to  an  equivalent  one  having  a  rational 
denominator. 

2     _       2  V3        _  2\/3  _2  a/3. 
5  a/3  ""  5  a/3  X  a/3  ""  5  a/32  ~"    15    ' 


2  a/5  _  2  a/5  X  V3  _  2  a/125  x  a/9  _  2  a/1125  ^ 
3V9~3V9xV3~        3V27         ~'      9        ' 


V^        a/^xV^^      V^xa/^"^      a/^"^" 


Hence,  denoting  the  degree  of  the  denominator  of  the  given 
fraction  by  n,  we  have  the  following 


RULE. 

Divide  some  perfect  n^^  power  which  is  a  multiple  of  the  quan- 
tity under  the  radical  sign  in  the  given  denomi^iator  by  that 
quantity,  and  multiply  both  terms  of  the  given  fraction  by  the 
indicated  n^  root  of  the  quotient. 


198 


BADICAL    QUANTITIES. 


HXAXPZES. 


Reduce  each  of  the  following  fractions  to  an  equivalent  one 
having  a  rational  denominator : 


Ans. 
Ans. 


3 

2V9 


2 

vr 
2 

w 

2a/6  4 


2 


6Va^ 


Ans. 


5.    ^=r.      -4n«. 

2V3a; 


8. 


10. 


11. 


V2 
V9' 

a/2 

V3 
V2" 

a/2 

vr 

a/2* 

V! 

V3' 


^W5. 


2Va 
5a2  ' 

flA/6 
2    ■ 

V72 


^W5.  V2. 


JW5. 

V41472 

2 

Arts. 
Ans. 

a/25  X  812 

3        • 

*a/288 
2     • 

Ans. 

'\/8  X  9^ 

13. 


13. 


Vb' 


Vb 


Vb 


15. 


16. 


2! 
J*' 

1 

a** 


-J" 


17-    4- 


18. 


19. 


20. 


21. 


a^/b 
b 

b 

n  n 7' 


aW 


22.    ^%. 


V. 


Ans. 


aVb 


,       aVW 
Ans.  — 5—. 
0 


b     ' 

Ans, 
Ans. 

ah^ 

b    ' 

1    m-l 

a^'b"' 
b      ' 

mn-^ 

Ans, 

ab   " 

Ans.  — , 
a 


Ans. 


Ans. 


a 

Vb 

a 


Ans. 


Ans, 


^/~ac 


KEDUCTION-.  199 

323.  To  reduce  a  fraction  whose  denominator  is  a 
binomial  surd  to  an  equivalent  one  having  a  rational 
denominator. 

1.  Reduce to  an  equivalent  fraction  having  a 

2  V3  -  V2 

rational  denominator. 

6  5  _  5(VT2  4-V2) 


2a/3  — V2       VT2-V2      (a/12  — V2)(Vl2  +  a/2) 

_  5(a/12  +  a/2)  _  a/12  +  \/2^ 
""         12  —  2        ~  2 

We  obtain  the  multiplier  VT2  -f  a/2  by  dividing  12  —  2  by 

\/T2  — a/2. 

5 

2.  Reduce  = —  to  an  equivalent  fraction  having  a 

2  a/3  +  a/2 
rational  denominator. 

5  5  5(a/T2— a/2) 


2  a/3  +  a/2   a/12  +  a/2   (vl2  +  a/2)  (vT2  —  a/2) 

_5  (vl2  —  a/2)__  a/12  — a/2 
"~   12  — 2   ~"    2 


3.  Reduce  5-^^ g— =  to  an  equivalent  fraction  having  a  ra- 

A/a  —  V^ 
tional  denominator. 

Dividing  a  —  J  by  V«  —  V^j  we  obtain  V^  +  a/«^  +  A^ 
Multiplying  both  terms  of  the  given  fraction  by  this  quotient,  we 
have 

c(a/^+a/^+  Vb^) 
a  —  b 

c 

4.  Reduce  57= 5/=  to  an  equivalent  fraction  having  a  ra* 

tional  denominator. 

Dividing  a  -f  5  by  V«  +  V^,  we  obtain  V?  —  Vad  +  \/^\ 


200  RADICAL    QUANTITIES. 


5.  Keduce  ^— : ~  to  an  equivalent  fraction  having  a  ro" 

V  «  —  yb 
tional  denominator. 

c        __        c         _         c{y^-^Vab  +  lVa  +  V¥)        _ 

We  reduce  the  given  fraction  to  an  equivalent  one,  in  which 
the  simple  surds  in  the  denominator  are  of  the  same  degree,  and 
then  multiply  both  terms  of  the  resulting  fraction  by  the  quotient 
obtained  by  dividing  a  —  ^  by  V^  —  V^. 


MULES. 

I.  If  the  given  denominator  is  of  the  form  of  V«  —  *Vb,  divide 
the  indicated  difference  of  the  quantities  under  its  radical  signs 
by  the  denominator,  and  multiply  both  terms  of  the  givm  fraction 
by  the  quotient, 

II.  If  the  given  denominator  is  of  the  form  of%^  +  V^?  f^nd 
its  indices  ere  even,  proceed  as  directed  in  L 

III.  If  the  given  denominator  is  of  the  forjn  of  ^fa  +  a/^,  and 
its  indices  are  odd,  divide  the  indicated  sum  of  the  quantities  un- 
der its  radical  signs  by  the  denominator,  and  multiply  both  terms 
of  the  given  fraction  by  the  quotient. 

IV.  If  the  given  denominator  is  not  of  the  form  of  V^  —  V^, 

nor  of  the  form  of  \fa  -\-  Wb,  reduce  the  given  fraction  to  an  equiv- 
alent one  having  a  denominator  of  one  or  the  other  of  these  forms 
(306),  and  proceed  with  the  result  as  directed  in  the  rule  which 
corresponds  to  the  form  of  its  denominator. 


REDUCTION.  201 

Cor.  1. — If  we  multiply  both  terms  of  the  fraction — =. 

a±  Vb 
by  a  =F  V^j  the  resulting  fraction  will  have  a  rational  denomina- 
tor; for 

a±Vl  =  V^  ±  Vh,     and    a^  —  b-r-  {V^  ±  Vb)  =  «  =F  V^ 

Cor.  2. — A  fraction  whose  denominator  is  a  trinomial  of  the 
form  of  Va  ±  Vb  ±  a/c  may  be  reduced  to  an  equivalent  one 
having  a  rational  denominator  by  two  multiplications.    Thus, 

d  _  d{Va±^TVc)  ____ 

d{Va±VhTV~c)  _  d{Va±VbTVc){a±i—cTWab)  __ 
a±b—c±2Vah   ~  {a±b—c±2Vab)  {a±b—c^^2Vab)~~ 

d{Va±VbTVc)(a±b—cT2Vab)^ 
ia±b—cy—^ab 

EXAMPLES, 

Reduce  each  of  the  following  fractions  to  an  equivalent  one 
having  a  rational  denominator: 

1.  — ^— .  Ans  !M±:v^) 

A/5-V2  3 

V5-A/2  ^ 

„   a—Vl                                                   .       a^+b—2aVb 
3. —  Ans, 5 — 7 — . 

a-^Vb  ^'-^ 

^a-\-^/b  \       a  +  b  +  2Vab 

5.  lr±^.  Ans.  '-±fi. 

3(3— 3  V2)  ^ 


202  RADICAL    QUANTITIES. 


6.  r-/-T7-  ^ns.  5  (V9  + V6  + Vi). 


Ans, 


d  (a^W-aWbc  +  acVb-V^) 


dVb+Vc  '  a^b—c^ 


^   Va-\-x-[-Va—x  .       a-\-Va^—^ 

9.  —-=^ »  -4ws. . 

Va+a;— ya— a;  ^ 

10.  ^^ — ■ ■ — ^— — ^^ ^.    Ans.  5^ — j ^. 

(x^+x-^iy-^(xi-x-iy  ^'^^ 

324.  Utility  of  tJie  Two  Preceding  Transforma- 
tions,— The  two  preceding  transformations  enable  us,  in  many 
cases,  to  abridge  the  computation  of  the  approximate  value  of  a 
numerical  fraction  whose  denominator  is  a  surd. 


JXi  USTJRATIONS. 

>v/5  4-V^_  V40+V24_2(\/l0  +  V6)_  VTo+Vq 
V8       "  8  -  8  "  4         • 

2.      i.^%,  =  TiVE-WlS)  ^  248024  ^  ^^^^^^ 

3^        V6       ^  V42-V18  ^  2^  ^  ^^^^^^^ 
V7  +  V3  ^  4 

.      9  +  2V1O       2(9  +  2vloy       242  +  72a/10       .  __ 
18-4\/l0  164  164 

2^/^      _2V3(V25-V30  +  V36)_ 
2(V75-V90  +  Vl08)       ^.^_ 


lERATIONAL    QUANTITIES.  203 

PROPOSITIONS  RELATING  TO  IRRATIONAL  QUANTITIES. 

325.  An  irrational  quantity  cannot  de  expressed  by  a  rational 
fraction. 

This  follows  from  the  definition  of  an  irrational  quantity 
(398). 

326.  A  simple  quadratic  surd  cannot  le  equal  to  the  sum  of 
a  rational  quantity  and  a  simple  quadratic  surd. 

For,  if  possible,  suppose 

V^  =z « -f-  Vm    .    .     .     (1), 
in  which    ^/n  and   Vm  are  surds. 
Squaring  both  members  of  (1), 

7i  =  ^2  _|_  2a  ^m  +  m : 


y2 


m 


whence,  ^m=i ^r .    .    .     (2); 

<»a 

that  is,  we  have   Vm,   an  irrational  quantity,  equal  to  a  rational 
fraction,  which  is  impossible  (325) ;  hence  (1)  cannot  be  true. 

327.  Tlie  product  of  two  simple  quadratic  surds,  which  are 
not  similar,  and  which  cannot  be  made  similar,  is  irrational. 

Let   Vm  and   Vn  be  two  such  surds;  then,  if  possible,  sup- 
pose 

Vm^  =.an    .    .     .     (1). 

Squaring  both  members  of  (1), 

mn  =  d^v?  \ 

whence,  \/m  =  a  Vn    .    .    .    (2) ; 

that  is,   Vm  and   Vn  may  be  so  reduced  as  to  be  similar.     But 
this  is  contrary  to  the  hypothesis ;  hence  (1)  cannot  be  true. 


204  RADICAL    QUANTITIES. 

328.  Tlie  quotient  of  itvo  simple  quadratic  surds,  which  are 
not  similar,  and  lohich  cannot  he  made  similar,  is  irrational. 

Let   Vrn  and   's/n  be  two  such  surds ;  then,  if  possible,  sup- 
pose _ 

4/f  =  a»    .    .    .    (1). 

Squaring  both  members  of  (1), 
in 


=  a*w 


2«3; 


•whence,  ^/mz=an's/n    .    .    .     (2); 

that  is,   "s/m,  and   Vn  may  be  made  similar.     But  this  is  con- 
trary to  the  hypothesis;  hence  (1)  cannot  be  true. 

329.  Tlie  sum,  or  difference  of  tivo  simple  quadratic  surds, 
which  are  not  similar,  and  which  cannot  he  made  similar,  cannot 
he  equal  to  a  simple  quadratic  surd. 

Let  Vrn  and  Vn  be  two  such  surds;  then,  if  possible, 
suppose 

Vrn  ±  Vn  =  Va    .    .    .     (1). 

Squaring  both  members  of  (1), 

m  ±  2  Vnm  -{-  nz=a; 

u                     .      / —      a  —  m  —  n  ... 

whence,  ±  yinn  = .    .    .     (2). 

But  Vmn  is  irrational  (327) ;  hence  we  have  an  irrational 
quantity  equal  to  a  rational  fraction,  which  is  impossible  (325) ; 
hence  (1)  cannot  be  trua 

330.  In  an  equation,  of  which  each  memher  is  the  sum  or 
difference  of  a  rational  quantity  and  a  simple  quadratic  surd,  the 
rational  quantities  of  the  two  memhers  are  equal,  and  also  the 
irrational  quantities. 

Suppose       X  ±  Vy  =  a  ±  Vh    .    .    .     (1), 


lERATIONAL    QUANTITIES.  205 

in  which    ^/y   and   V^    are  irrational ;    then  will  x-=za  and 
V^  =  V^.    For  suppose 

ic  =  «  ±  w    .    .    .     (3) ; 
then  (1)  becomes 

whence,  n  ±,  ^fy  =  ±  Vb    .    .    .     (3). 

But  (3)   is  impossible   (326) ;    hence   (2)   cannot  be  true. 
Therefore  x  =za,  and  consequently   Vy  =  V^. 


331.  If  i/a  -{-  Vb  =zx  -{•  A^y,  in  which   ^/b  and  V^  are 

irrational,  then  ya  —  V^  =  a;  —  Vy. 
For  since 

|/a  +  V^  =  a;  +  a/^    .    .    .    (1), 

we  have  by  squaring, 

a  +  Vb  =  x^^-2xVy-\-y   .    .   .    (3); 

.-.     «  =  2:2  +  y  . . .  (3),    and     V^  =  2a;  a/^  . . .  (4)  (330). 
Subtracting  (4)  from  (3), 

a  —  Vh  =zx^  —  2x  Vy  +  y    .    .    .    (5) ; 


whence,  y  a  —  Vb  =:x  —  Vy    .    .    .    (6). 

333.  7/"  y  a  4-^/3  =  a/^  +  Vy ,  in  which  Vb,  ^/x,  and 
Vy  are  irrational,  then  y  a—  Vb  =  Vx—  Vy- 
For  since 


|/a  +  V^  =  Va;  +  Vy    •    •    •     (1)> 


206                                      EADICAL    QUANTITIES. 

we  have  by  squaring, 

a-\-  Vl  =  x-{-2  Vxy  H-  y    .    . 

.    (2); 

.-.     a  =  a;4-y ...  (3),    and     Vb=2Vxy, 

..(4)  (330). 

Subtracting  (4)  from  (3), 

a—Vb  =  x  —  2Vxy-^y    .    . 

.   (5); 

whence,         ya  —  Vl  =  Vx—Vy    .    .    . 

(6). 

SIMPLIFICATION  OF  COMPLEX  RADICAL  QUANTITIES. 

333.  A  Conijilex  Radical  Quantity  is  an  expres- 
eion  in  which  one  radical  sign  includes  one  or  more  others.   Thus, 

yVS,  |/9  4-3v^,   and  \  0A/hy~Q  are  complex  radical  quan- 
tities. 

334.  The  complex  radical  quantity  y  a  ±  ^/h  may  le  sim- 
plified if  b  is  a  perfect  square,  or  if  a^  —  h  is  a  perfect  square. 

1.  Suppose  that  5  is  a  perfect  square;  then  y  d±:Vb  may- 
be reduced  to  Va±Cy  in  which  c  is  the  square  root  of  h.  Thus, 
|/5±\/9  =  V'5±3. 

2.  Suppose  that  "s/b  is  a  surd,  and  that  a^  —  h  is  a  perfect 
square;    then     ya-\-Vb    may  be  reduced  to     ±  y  ^"^^  J_ 


iA~^j  and  \/a-Wb  may  be  reduced  to  ±  i/^±£  T  j/^""^ 
r      2  ^  V      2         '^      2   ' 

in  which  c  is  the  square  root  of  a^  —  b. 


Assume      Vx  -\-  Vy=A/a-{-Vb    .    .    .    (1). 

In  this  equation  one  or  both  of  the  terms  in  the  first  member 
must  be  irrational,  because  the  second  member  is  a  surd ; 


SIMPLIFICATION.  207 

V^-^Vy=\/a-Vb    .    .    .     (2) (331-332). 

Multiplying  (1)  by  (2), 

x-y=V'^^b    .    .     .     (3). 
Squaring  both  members  of  (1), 

a;  +  2  Vx^  +  ^  =  «  +  VJ    .     .    .     (4). 
In  this  equation  2  Vxy  is  a  surd  (327) ; 
x-\-yz=a    .    .    .    (5). 
Combining  (3)  and  (5),  we  find 


2 


(6), 


and  Vy=  ±  j/a—Va^—b  r,^>. 


(8), 


/^Zvi=±v^+^~^'-^:fV^^^'-^ , , . 


But  Va^—  b=c  by  hypothesis; 


|/a+V^=±/^±y^    .     .     .     (10), 


and       ^a^Vb=±  |/^  ip  >j/^     .    .    .    (11). 


EXAMPIjES. 


1.  Simplify  |/3  +  2\/2. 


|/3  +  2V2=|/3  +  V8; 


208  RADICAL    QUANTITIES. 


a  =  3,    ^>  =  8,    and    c  =  a/9— 8  =  1. 
Substituting  in  (10), 


i/3  +  2a/2  =  ±|/S  ±  |/?Z^  =  ±  V2  ±  1. 


2.  Simplify  |/7-2VlO. 


|/7-2a/10  =  |/7-\/40; 


a  =  7,    J  =  40,    and    c  =  V49-40  =  3. 
Substituting  in  (11), 


-j/7-2Vl0  =:±V6TV2. 
Simplify  each  of  the  following  expressions : 

3.  |/i1  +  6a/2.  Ans.  ±3±\/2. 

4.  y/7_4V^.  Ans.  ±2^V3, 


5.    1/94+42 a/5.  ^w5.  ±7±3a/5. 


6.    |/ll  +  6 V'2  +  |/7-2a/10.  ^7i5.  ±3±a/5. 


7.    4/Jc  +  2Z>a/^c— R  ^,i5.  ±5±a/&c— Z>2. 


8.    |/(a  +  J)2— 4(«— J)  a/«^.  ^^.  ±(a— &)=F2A/a?. 


335.  Tlie  complex  radical  quantity  y  a\fc-^Vi^  in  which 
aVc  and  a/^  are  supposed  to  le  surds,  may  he  simplified^  if 
fl2 {g  d  perfect  square. 


SIMPLIFICATION^.  200 

aVc  ±  Vb  =  Vela  ±  y -jf 


'''^aVc±^b  =  VV~((a±\/-]=  ^W a±\/\  (302). 

This  expression  may  now  be  simplified  by  the  method  of  Art. 
334  when  c^ is  a  perfect  square. 


EXAMPLJES, 


1.  Simplify  |/V324VM. 

a/32  +  V30 = V2  (4  +  a/15)  ; 


|/a/32  +  V'SO  z=  V2>(/4+  Vl5. 


But  /4+A/r5=±/|±|/^. 

Simplify  each  of  the  following  expressions. 

2.  |/a/27  +  Vl5.  ^ W5.  V3  f  ±  ;^  ±  |/i  j. 

3.  |/5a/2  +  4a/3.  ^ws.  V2(±a/3±a/2). 


4.     |/8a/3~2a/45.  ^^.  a/3(±a/5  =F  V3). 

"  /       ^ 
336.  Complex  radical  quantities  of  the  form  of    y  a\/b 

may  ie  simplified  by  the  rules  of  Art,  318.     Thus, 

"sa/B  =  yV^  =  V45. 


14 


210  RADICAL    QUANTITIES. 

337.  Complex  radical  quantities  of  the  form  of 

3 


iUayi)  ±  c^d  ±  etc.,  or  of  the  form  of  y  a\/b  ±  c  \/5  ±  etc., 
may,  in  some  cases,  be  simplified  by  the  method  of  Art,  319.  Thus, 
i/ V5  +  2  Vl5  +  a/3  =  V5  H-  V3. 

IMAGINARY  QUANTITIES. 

338.  An  Imaginary  Quantity  is  one  which,  when  in 
its  simplest  form,  contains  an  indicated  even  root  of  a  negative 

quantity.  Thus,  2  V^,  5  V— 10,  and  (a  +  i)  V^  are 
imaginary. 

339.  The  term  Meal  is  applied  to  all  quantities  that  are  not 
imaginary.    Thus,  5,  —  3,  VS,  and  V— 27  are  real 

340.  Imaginary  quantities  are  classified  in  the  same  way  as 
other  surd  quantities.    Thus,  2  V— 3  is  simple  and  of  the  second 

degree,  4/3  +  2  \/^  is  complex,  and  8  +  3  V^  +  5  V^i— 

7  V — 1>  considered  as  a  single  expression,  is  a  compound  or 
polynomial  imaginary  quantity. 

An  imaginary  quantity  usually  consists  of  a  real  and  an  imag- 
inary part.  Thus,  2  +  3  V— 1  consists  of  the  real  part  2  and 
the  imaginary  part  3  V— 1.  The  whole  expression  is  considered 
as  an  imaginary  quantity  on  account  of  the  presence  of  the  imagi- 
nary part. 

COMBDfATIONS    OF    SIMPLE    IMAGINARY    QUANTITIES. 

341.  To  find  the  sum  of  simple  imaginary  quan- 
tities. 

EXJLMrLES. 

1.  Find  the  sum  of  V-^  and   V— 16. 
>v/I~9  +  V-IQ  =  VQ  (-  1)  +  V16(-1)  =  3  \/^ 


IMAGIKARY    QUANTITIES.  211 

2.  Find  the  sum  of  3  V— 81   and  2  V^16. 


3  V^ZM  +  2  V-16  =  3  V81(-l)  +  2  Vl6(-1)  = 

9  v^^  +  4  V^i  =  13  V^. 

3.  Find  the  sum  of   V— 50  and   V— 18.       ^W5.  8  V^^. 

4.  Find  the  sum  oi  ^/  —a  and   ^  —h. 

Ans.  (Va  +  Vb)  a/^I. 

342.  To  find  the  diflference  of  two  simple   imag- 
inary quantities. 

EXAMPLES, 

1.  Subtract  2  V^^  from  9  V^I 

9  V^^  -  2  V^^  =  9  V^l  —  2  V4(— 1)  =  9  V^Ti— 
4  V^i  =  5  a/^i. 

2.  Subtract  2  V^^   from  9 


9  v^-  2-2  V—  3  =  9  V2(-l)  -  2  a/3(— 1)  = 
9  \/2  V^iri  _  2  a/3  a/^I  =  (9  a/2  —  2  a/s)  a/^I. 

3.  Subtract   V^^  from   V— 8^.  Ans.  V-^- 

4.  Subtract   V—  b  from   V~—  a, 

Ans,  (Va  —  Vi)  V—  1. 

343.  To  find  the  product  of  two  simple  imaginary 
quantities  of  the  second  degree. 

£!XAMPZES. 

1.  Find  the  product  of  b  V—  ci  and   c  V—  «• 

It  is  evident  that  b  ^/—a  x  c  \/~^a  =  bc{—a)  (314,  CoE.  1) 
=  —  abc.    It  is  also  evident  that  b  V  —  «  X  c  a/—  a=.bc  Va^; 


212  RADICAL    QUANTITIES. 

for  if  this  be  not  true,  the  rule  for  the  sign  of  a  product  is  not 
general ;  it  therefore  follows  that,  in  this  case,   V«^  =  —  a. 

But  it  may  be  said  that  V«^  =  ^y  and  therefore  a  =  —  a. 
This  reasoning  is  erroneous,  for  it  is  not  true  that  v^  =i  -\-  a 
and   —  a  at  the  same  time  (74). 

We  are  enabled  to  remove  the  ambiguity  with  regard  to  the 

sign  of  a/«^  by  knowing  that  a^  resulted  from  the  involution  of 
—  a.  If  we  did  not  know  in  what  way  a^  was  produced,  that  is, 
whether  a^  represented  {-\- of  or  (— a)^  then  the  sign  of 
V^  would  be  ambiguous. 

2.  Find  the  product  of  a/—  a  and   ^/—b, 

a/^a  =  ^/a{—l)  =  Va  ^/~—  1, 
and  V^^  =  V^*  (— 1)  =  V^\/^^; 

.•.    a/^^  X  V— *  =  Va  ^f^^  X  ^/h  V—  1  =  VabW—  l) 
=  —  a/«^. 

The  ambiguity  with  regard  to  the  sign  of  the  product  may 
therefore  be  removed,  if  we  reduce  each  of  the  imaginary  factors 
to  the  form  of  a  V— 1?  and  remember  that   V— Ix  V— 1  or 

/   / \^  • 

\V  —  1)    IS  equal  to    —  1. 

3.  Multiply  4  ^^l  by  3  a/^^.  Ans.  —  12  Vs. 

4.  Multiply   —  5  a/^2  by   —  3  a/^^.     Ans.  —  15  a/10. 

5.  Multiply   a/— ^«^  by   ^T^W.  Ans.  —  ah. 

344.  To  find  the  quotient  of  two  simple  imaginary 
quantities  of  the  same  degree. 

JEXAMTZJES. 

I.  Divide   \^—a  by  V—  h. 

V—a  _  VaV^l^  _  Va  _     /^ 


IMAGIJS^AR'Y    QUANTITIES. 


%13 


2.  Divide 


V~~  a  hj   —  V—  i 


h  a/_  h  V    h' 


3.  Divide   a/—  a  by   —  V—  b. 


—  ^—i     —  v^V-1 


v& 


=-1^' 


The  ambiguity  with  regard  to  the  sign  of  the  quotient  of  two 
imaginary  quantities  is  removed,  therefore,  by  reducing  each  of 

- 1,  and  observing  that  — ^=r  =  1. 


V-i 


Ans,  ^\/3. 


Ans, 


eVs 


them  to  the  form  of  « '^ 

4.  Divide   6  V-^  by  2  v^^^. 

5.  Di^dde    —  V^^  by   —  6  V~—3, 
345.  To  find  all  the  powers  of  V— 1. 

(V:riy  =  v--^, 
(V:ri)'^_a, 

(V--ri/  =  (-if^i. 

If  we  multiply  these  powers,  in  their  order,  by  the  4th,  we 
shall  obtain  the  5th,  6th,  7th,  and  8th ; 

(V--ri)'^_i, 

/     / \8 


214  RADICAL    QUANTITIES. 

Therefore  all  the  powers  of  ^—  1,  arranged  in  order,  be- 
ginning with  the  lowest,  form  a  repeating  cycle  of  the  following 
terms:     V— 1,   —1,    —  V— 1?  audi. 

346.     MISCELLANEOUS  EXAMPLES  IN  IMAGINARY  QUANTITIES. 

If  the  student  will  observe  the  directions  given  in  Articles 
343  and  344,  and  remember  that  imaginary  quantities  are  surds, 
he  will  have  no  difficulty  in  solving  the  following  problems : 

1.  Multiply  4  +  \/^^  by  V^^- 

Ans,  4  V— 5  —  a/15. 

2.  Multiply  3  +  V^^  by  2  —  ^T^^. 

Ans.  6  +  2  V--2  —  3  V'^^  +  VS. 

3.  Multiply   1  +  V^^  by  I  —  V^^.  Ans.  2. 

4.  Multiply  a-\-h  V^^  hy  a  —  b  V^^.     Ans.  a^  +  h\ 

5.  Divide   (\/^i)*  by  V^^.  Ans.  —  V^^. 

6.  Divide  4  +  V"^  by  2  —  V^^.       ^W5.  1  +  ^/^2. 

7.  Reduce to  an  equivalent  fraction  having  a  ra- 
tional denominator.  Ans.  V^l. 


8.  Simplify  y^7  +  30  V—  2.  ^W5.  ±  5  ±  3  ^^^2. 


9.  Simplify  /j/si  +  12  V^^  +  |/-  1  +  4  V-  5. 

^W5.  ±8±2V^5. 


10.  Simplify  y^a^  _  2a^  +  2  (a  —  h)  V— ^. 

Jws.  ±  («  —  i)  ±  J  a/^^. 

11.  Find  the  3d  power  of  a  a/— •  1.  -4;i«.  —  fl^A/--^i. 


RADICAL    EQUATIONS.  215 

12.  Find  the  3d  power  of  «  —  Z>  V'^- 

Ans.  «3  4-  J3  >v/:^i  _  Sab{h  +  a  \^-^). 

13.  Find  the  4th  power  of  a  +  V^b. 

Ans.  a*  —  Qa^b  -\- b^  +  (4«8  _  4«5)  V^- 

14.  Find  the  values  of  x  and  «/  in  the  equation 

^  +  «/  +  ^  V^5  =  5  +  a;  +  2/  V^^- 

x  =  2  +  VlO, 


y  =  6  -\-  Via 

RADICAL   EQUATIONS. 

347.  A  Madical  Equation  is  one  which  involves  one 
or  more  radical  quantities. 

348.  To  free  a  radical  equation  from  radical  quan- 
tities. 

EXAMPLES, 

1.  Free  the  equation 

0/^-^3=2    ...    (1) 
from  radical  quantities. 

Transposing  V3  to  the  second  member,  and  squaring  the  re- 
sulting equation, 

a;  =  44.4-v/3 +  3  =  7  +  4^3    .    .    .     (2). 

Transposing  7  in  (2)  to  the  first  member,  and  squaring  the  re- 
sulting equation, 

je2_i4^4.49-48; 

whence,  dfi  —  14a;  =  —  1    .    .    .    (3). 

2.  Free  the  equation 

Va;  +  ll  +  \/^"^^  =  5    ...    (1) 
from  radical  quantities,  and  find  the  value  of  x. 


216  EADICAL    QUANTITIES. 


Transposing  Vrc  — 4  to  the  second  member,  and  squaring 
the  resulting  equation, 


a;  +  11  =  25  —  10  Va;  —  4  +  a;  —  4 ; 


whence,  V^c  —  4  =  1    .    .    .     (2). 

Squaring  (2),    a;  —  4  =  1 ;    whence,    x  =  6. 
3.  Free  the  equation 


/y/i  _  Va;  —  6  _  4a;  —  35  ,  . 

Vx  +  Va;  —  5  ^ 

from  radical  quantities,  and  find  the  value  of  x. 


Vx—Vx  —  5  _  2a;  —  5  —  2  Va^  —  5x  ,323) . 


Vi  +  Vx  —  5 
hence  (1)  becomes 


2a;  —  5  —  2  ViB^  —  5a;  =  4a;  —  35; 


(1) 


whence,  v^ —  5a;  =  15  —  a;    .    .    .    (2). 

Squaring  (2),    a;^  _  5^; :_  225  —  30a;  +  a?*; 
whence,  a;  =  9. 

4.  Free  the  equation 

c  m  ^/a ^ 

V^+  V^      a;  — a—Vi— Va 

from  radical  quantities,  and  find  the  value  of  x. 
Multiplying  both  members  of  (1)  by  x  —  a, 

c  {Vx  —  ^/a)  +  m  Va  =  m  ( Vi  +  v^)  > 

whence,  {c  —  m)  Vx  =  c  Va    .    .    .     (2). 

Squaring  (2),  (c  ■—  m)H  =  ac* ; 

whence,  x  =  -, r^ . 


RADICAL    EQUATIONS.  217 

Find  the  value  of  x  in  each  of  the  following  equations : 


5.     V^  +  7  +  v^  =  7.  Ans,  a:  =  9. 


G.    x  +  Z  =  Vx^  —  4:X  +  59.  Ans.  x  =  5. 

7.    \/Vx  4-  48  —  a/^  =  V^.  -4/iS.  a;  =  16. 


a^—4a 


4 


8.  a/x  +  2  a/«  +  a;  =  ya  —  V«  +  ^z^-    -4/is.  a: 

9.  —  +  —  =  4/ -.  -4/15.  a;  =  c  (Vm  —  «). 


10.  H-     ,  =     ^  Ans.x  =  -. 

Vi-{-x     Vi—3?^     Vi—s^  ^ 


11.     V  ^  +  2''  =     ,  ^ws.  a;  =  — tr — . 

c2  — a2 


12.    g  +  ^/&  —  (ix=  -— .  Ans,x  = 

\(^  —  ax  ^ 


1^-     S  +  5  =  ^  25  +  5^5  +  ^-  ^^^.^  =  20. 

14.    Va  —  a:  =  V«  +^'  ^^i^*  a;  =  — ^ — . 


^2-a/S         V4  +  a;  13 

16.  V5  +  ^  +  Vs"^^  =  VlO.  Ans.  x  =  5. 

17.  yo;  +  ya  +  a;  =1  -.  Ans.  x  =  -. 

Va  -\-  X  ^ 


18 


.    X  -\-  a  =  jJ €?  4-  a;  V^  +  a?^.  -4ws.  a;  = 


^-4^2 

4a     • 


218  RADICAL    QUANTITIES. 


19. 


V^x  +  'Z       4^/6^+6* 
4  +  x 


20.     V64  H-  a:^  -  8a;  =  ? 


21.    V5  +  »  +  VS 


V4  +  a;' 
15 


VS  4-aJ 


^W5.  a;  =  6. 

^W5.  ir  =  3. 
^W5.  iC  =  4. 


!.    ^x+Vi-  ^x-^x  =  K^rp^)  .    An,.  xJ±. 


^/ax  —  h      3Vax  —  2h 

A6.       — -=. =  -:s=. . 

\ax  -\-h      3  wax  4-  5 J 


24.     V4a:  +  1  + V4^^g^ 


25. 


V4x  H-  1  —  \/4^ 

3V^^4_3V^-f.l5 
V^4-2""     V«  +  40 


9^2 

a 


x  = 


Ans.  a;  =  4. 


349. 


EH 

i 


Definitions  . . 


SYNOPSIS    FOR    REVIEW. 


'  Simple  radical  quantity. 

Radical  factor  and  its  coefficient. 

Degree  of  dmple  radical  quantity. 

Similar  radical  quantities. 

Simplest  form  of  radical  quantity. 

Rational  quantity. 
L  Irrational  quantity. 


Reduction.  . 


'  To  reduce  rational  quantity  to  radical  quantity  of 

n'*  degree.    Rule. 
To  introdu^  coefficient  of  radical  factor  under 

radical  sign.    Rule. 
To  remme  a  factor  from  binder  the  radical  sign  to 

the  coefficient.     Rule. 
To  reduce  the  indicated  root  of  a  fraction  to  an 

equivalent  ea-pressioji  in  which  the  quantity 

under  the  radical  sign  sMU  he  entire.    Rules. 

Cor. 


SYNOPSIS    FOR    REVIEW. 


219 


SYNOPSIS    FOR    REVIEW— Continued. 


Reduction— (7(cmf(f. 


Combinations  .  . 


Involution  op  Rad- 
ical Quantities. 


Evolution  op  Rad- 
ical Quantities. 


Reduction  of  pkac- 
tions  having  surd 
denominators  to 
equivalent    ones 

HAVING     rational 

denominators. 


To  reduce  simple  radical  quantity  to  sim- 
plest form.    Rules. 

To  reduce  radical  quantity  of  the  form  of  ^^a* 
to  another  of  lower  degree.     Rule. 

To  reduce  simple  radical  quantity  to  another 
of  higher  or  lower  degree.  Rule.    Cor.  1,  3. 

To  reduce  simple  radical  quantities  liaving  un- 
equal indices  to  equivalent  ones  having 
equal  indices.     Rule. 

To  find  the  sum  of  sim,ple  radical  quantities. 
Rules. 

To  find  the  difference  of  two  simple  radical 
quantities.    Rules. 

To  find  the  product  of  two  or  more  simple  rad- 
ical quantities.    Rules.     Cor.  1,  2. 

To  find  the  product  of  polynomial  radical 
quantities. 

To  find  the  quotient  of  two  simple  radical 
quantities.    Rules.    Cor.  1,  2. 

To  find  the  quotient  of  polynomial  radical 
quantities. 

"  To  raise  the  indicated  nP"  root  of  a  quantity  to 
any  power.    Rule.    Cor.  1,  2,  3. 

To  raise  a  simple  radical  quantity  to  any 
power.    Rule.    Cor. 

To  raise  a  polynomial  radical  quantity  to  any 
power. 

To  find  any  root  of  the  indicated  root  of  a 

quantity.    Rules.    Cor. 
To  find  any  root  of  a  simple  radical  quantity. 

Rules.     Cor. 
To  find  the  square  root  or  cube  root  of  a  poly- 

nomial  radical  quantity. 
A  simple  surd. 
A  polynomial  surd. 
To  reduce  a  fraction  whose  denominator  is  a 

simple  surd  to  an  equicalent  one  having  a 

rational  denominator.    Rule. 
To  reduce  a  fraction  whose  denominator  is  a 

binomial  surd  to  an  equivalent  one  having 

a  rational  denominator.    Rules.    Cor,  1, 2. 
Utility  of  preceding  transformations. 


220 


a 

I 

5 


RADICAL    QUANTITIES. 
SYNOPSIS   FOR   B^YJBW— Continued, 


Prop,  relating 
TO  Irrational  < 
Quantities. 


325. 
326. 
327. 
32§. 
329. 
330. 
331. 
L332. 


'  A  complex  radical  quantity. 


Simplipica'n  op 
Complex  Radi- 
cal Quan, 


To  simplify 


Imaginary 
Quantities. 


\/a±Vb, 

ya  V^  ±  cVd  ±  etc. 
/j/a  V^  ±  c  Vd ±  etc. 


'  An  imaginary  quantity. 
A  real  quantity. 
Classification  ofimxiginary  quantities. 

To  find  the  sum. 
To  find  the  difference. 
To  find  the  product. 
To  find  the  quotient. 

.  To  find  all  the  powers  of  y  —  1. 


Combinations  of 
imagiTiary  quan- 
tities. 


CHAPTER    XIII. 

QUADRATIC  EQUATIONS  WITH  ONE  UNKNOWN  QUANTITL- 
QUADRATIC  EXPRESSIONS. 


DEFINITIONS  AND  PRINCIPLES. 

350.  An  equation  which  contains  only  one  unknown  quan- 
tity as  X,  and  whose  members  are  entire  and  rational  with  refer- 
ence to  X,  is  of  the  Second  Degree  when  it  contains  a^  and 
does  not  contain  a  higher  power  of  x.     Thus, 

7x^  =  dx  +  160 

is  an  equation  of  the  second  degree. 

351.  A  Quadratic  Equation  is  an  equation  of  the 
second  degree. 

352.  An  equation  of  the  second  degree  containing  only  one 
unknown  quantity  as  x,  when  expressed  in  such  a  form  that  its 
members  are  entire  and  rational  with  reference  to  x,  cannot  have 
more  than  three  kinds  of  terms,  namely :  terms  which  contain  the 
square  of  x,  terms  which  contain  its  first  power,  and  known  terms, 
that  is,  terms  independent  of  x.  Therefore,  by  transposing  and 
uniting  terms,  the  equation  can  be  made  to  take  the  form  of 

aoi?  •\-hx=:c'', 

a,  h,  and  c  being  given  quantities,  which  may  be  either  positive  or 
negative.    For  example,  the  equation 

can  be  transformed  successively  into  the  following  equations : 


222  QUADRATIC    EQUATIONS. 

6«2  _  30^:2  ^  135^  +  78a;  =  360  +  18, 
—  25a^  H-  213a;  =  378, 
25a:2  _  213a:  =  -  378. 

We  may  consider  a  in  the  general  equation  as  positive ;  for,  if 
it  is  negative,  we  may  make  it  positive  by  changing  the  signs  of 
all  the  terms  of  the  equation,  as  in  the  preceding  example. 

353.  A  Comjyfete  Equation  of  the  Second  De- 
gree is  one  which  can  be  expressed  in  the  form  of 

ax^  -\-  bx  =  c, 

in  which  neither  b  nor  c  is  zero.  Thus,  a;^  -f  5a;  =  24  and 
2a:2  —  3a;  =  2a;  +  12  are  complete  equations  of  the  second  degree. 

The  coefficient  a  cannot  be  zero ;  for  then  the  equation  would 
cease  to  be  of  the  second  degree. 

A  complete  equation  of  the  second  degree  is  sometimes  called 
an  Affected  Quadratic  Equation, 

354.  If  J  or  c  is  zero,  the  equation  takes  one  of  the  forms 

ax^  =  c,    aa^  -\-  hx  =^0. 

In  either  case  the  equation  is  said  to  be  Incomplete,  Thus, 
dx?  =  27  and  2x^  —  6a;  =  0  are  incomplete  equations  of  the 
second  degree. 

An  incomplete  equation  of  the  second  degree  of  the  form  of 
ax*'  =  c  is  sometimes  called  a  Pure  Quadratic  Equation. 

INCOMPLETE  EQUATIONS  OF  THE  SECOND  DEGREE. 

355.  To  solve  an  equation  of  the  form  of  ax?  =  c. 

Dividing  both  members  of  the  equation  by  a,  and  extracting 
the  square  root  of  both  members  of  the  resulting  equation,  we  find 


•1 


INCOMPLETE    EQUATIONS.  223 

R  ULE. 

Find  the  value  of  the  square  of  the  unlcnown  quantity  ly  the 
rule  for  solving  a  simple  equation  ;  the  result  will  be  an  equation 
of  the  form  of  x^  =  q  ;  then  extract  the  square  root  of  both  mem- 
bers of  this  equation. 

Cor.— The  two  roots  of  a  pure  quadratic  equation  have  equals 
ahsolute  values,  but  contrary  signs. 

EXAMPLES, 

Solve  the  following  equations : 


1. 

8                3-39      5^. 

-4/15.  a;  =  ±  3. 

2. 

{  =  U-3.». 

^^5.  a;  =  i:  2. 

3. 

^.-^-'X  16. 

Ans.  a;  =  ±  3. 

4. 

(a;  +  2)2  =  4:c  +  5. 

^ws.  a;  =  ±  !• 

5. 

1  -h  a;   '   1  —  a; 

-4^15.  X=i  ±-, 

6. 

3          1         7 
42^2       6a;2~3* 

Ans.  ^  =  ±  2' 

7. 

o     ,   7       65a; 

^ws.  a;  =  ±  2f 

8. 

>y/flr2  ^  x^  -\-X        h 

^a^  j^x^  —  x      c 

.                    a(b  —  c) 
Ans.  x=  ±  —^ — — ^. 

2Vbc 

Ans.  x=  ±  -4 '-. 

V2n-1 

9 

^,1     ^„2    1     ^2_          ^^ 

Va2  +  x^ 

0 

Va2-a;2_  V'52  4.a;2        c 

Va3  — a?J+ V^'^+ic^      ^ 

Ans.x-±y  2i(^+W) 


224  QUADRATIC    EQUATIONS. 

The  foUowiDg  equations  have  such  a  form  that  they  may  be 
solved  by  a  method  similar  to  that  employed  in  the  solution  of 
equations  of  the  form  of  a^  =  c : 

28a:     _  63  (g  +  18)  .. 

a;  4-  18  ~         ^x  '    '    '     ^  ^' 

^  Clearing  of  fractions,  (1)  becomes 

112ic2  =  63  (a:  +  18)2    .    .    .     (2). 

Dividing  both  members  of  (2)  by  7, 

16a:8  =9(x-\-  18)2    .    .    .     (3). 

Extracting  the  square  root  of  both  members  of  (3), 

4a:=±3(a:+18)     .    .    .     (4); 

whence,  a:  =  54    or    —  74- 

12.     (x  —  ay  =  h.  Ans.  x  =  a  ±  Vb, 

356.  To  solve  an  equation  of  the  form  of  ax^  +  ix  =  0. 
This  equation  may  be  expressed  thus : 

x{ax-\-b)  =  0    .    .     .     (1). 
Now,  in  order  that  the  product  of  x  and  ax  -]-  b  may  be  equal 


to  zero,  we  must  have 

either                            x  =  0    . 

.    .     (2), 

or                               era;  -f-  J  =  0 

.    .    .     (3). 

From  (3),                         x  = 

_b 
a' 

Hence,  an  equation  of  the  form  of   ax^  -\-  bx  =  0  has  two 
roots,  one  of  which  is  zero. 

EXAMPLES. 

Solve  the  following  equations: 

1.     2a^  -  ^  =  S^a:.  Ans.  x  =  0    or    9. 


PROBLEMS. 

225 

2. 

^35 

^-r  =  2^- 

Ans.  x  =  0    or    4. 

3. 

2  +  4-?  =  0. 

4. 

a;(2a;  +  5)=a;(3a;  — 9). 

357,  PROBLEMS, 

1.  Find  two  numbers,  one  of  which  is  four  times  the  other, 
and  the  sum  of  whose  squares  is  153.  Ans,  ±  3  and  ±  12. 

2.  Find  two  numbers,  one  of  which  is  three  times  the  other, 
and  the  difference  of  whose  squares  is  32.     A7is.  ±  2  and  ±  6. 

3.  Find  two  numbers,  one  of  which  is  three  times  as  great  as 
the  other,  and  whose  product  is  75.  Ans,  ±5  and  ±  15. 

4.  A  merchant  bought  two  pieces  of  cloth,  which  together 

measured  36  yards.     Each  piece  cost  as  many  dimes  a  yard  as 

there  were  yards  in  the  piece,  and  the  entire  cost  of  one  piece  was 

four  times  that  of  the  other.    How  many  yards  were  there  in 

each  piece  ? 

Ans.  24  yds.  in  one,  and  12  yds.  in  the  other. 

The  negative  numbers  are  not  given  because  they  do  not  sat- 
isfy the  question  in  its  arithmetical  sense. 

5.  Two  persons,  A  and  B,  set  out  from  different  places  to  meet 
each  other.  They  started  at  the  same  time,  and  traveled  on  the 
direct  road  between  the  two  places.  On  meeting,  it  appeared  that 
A  had  traveled  18  miles  more  than  B ;  and  that  A  could  have 
traveled  B's  distance  in  15|-  days,  but  that  B  would  have  been  28 
days  in  travehng  A's  distance.  Find  the  distance  between  the 
two  places.  Ans.  126  miles. 

6.  The  product  of  the  sum  and  difference  of  two  numbers  is  8, 
and  the  product  of  the  sum  of  their  squares  and  the  difference  of 
their  squares  is  80.     What  are  the  numbers  ? 

Ans.  ±  1  and  ±  3. 
16 


226  QUADRATIC    EQUATIONS. 

7.  The  product  of  the  sum  and  difference  of  two  numbers  is  a, 
and  the  product  of  the  sum  of  their  squares  and  the  difference  of 
their  squares  is  ma.    What  are  the  numbers  ? 


Ans.  ±y—^-    and     ±  |/ _— — • 


8.  Two  workmen,  A  and  B,  were  engaged  to  work  for  a  certain 
number  of  days  at  different  wages.  At  the  end  of  the  time,  A, 
who  had  been  idle  a  of  those  days,  received  m  dollars,  and  B,  who 
had  been  idle  b  of  those  days,  received  w  dollars.  Now,  if  B  had 
been  idle  a  days,  and  A  had  been  idle  b  days,  they  would  have  re- 
ceived equal  amounts.     For  how  many  days  were  they  engaged  ? 

b  Vm  —  a  Vn  , 

Ans.  — -=z —  days. 

ym —    Vn 


COMPLETE  EQUATIONS  OF  THE  SECOND  DEGREE. 

358.  To  solve  a  complete  equation  of  the    second 
degree. 

Let  us  consider  the  complete  equation 

acp^  +  bxz=zc    .    .    .     (1). 
Dividing  both  members  of  (1)  by  a, 

x^  +  h  =  -    .    .     .     (2). 
a        a  ^  ^ 

b  c 

Let  o  =  - ,  and  o  =  -',  then  (2)  becomes 
■^      a  ^      a  ^  ' 

x^^px  =  q    .    .    .     (3). 

Adding  ^  ^  ^^^^  members  of  (3), 

^^-px^^^q-V^    .    .    .     (4). 

The  first  member  of  (4)  is  the  square  of  (^  +  f ) ;  hence,  ex- 
tracting the  square  root  of  both  members, 


COMPLETE    EQUATIONS. 

'+1 

=  ±l/?+^     .     . 

Transposing  |, 

x.=  - 

-|±l/7^     •     . 

237 
(5). 


The  given  equation,  therefore,  has  two  roots,  namely : 


-i+/,.A 


The  operation  of  transforming  (3)  into  (4)  is  called  Completing 
the  Square, 

RULE, 

I.  Reduce  the  given  equation  to  the  form  of  x^  -\-px^  q. 

II.  Add  to  loth  memhers  of  this  equation  the  square  of  half 
the  coefficient  of  x. 

III.  Extract  the  square  root  of  both  members  of  the  equation 
thus  obtained ;  the  result  will  be  an  equation  of  the  form  of 

X  +  -^z=z  ±m,  from  which  the  values  ofx  may  be  found  by  trans- 
position, 

BXAMPZJES, 

Solve  the  following  equations : 

1  —  3x2  +  36a;  =  105. 

Dividing  both  members  by  —  3, 

ic2  _  12a;  =  —  35. 

Completing  the  square, 

ai2— 12a;+36=:-35  +  36=l. 


228  QUADRATIC    EQUATIONS. 

Extracting  the  square  root  of  both  members, 

x-6=±l; 

x  =  6±l; 

that  is,  ic  =  7  or  5. 

Verification.     -  3  x  7^  4-  36  x  7  =  -  147  +  252  =  105; 
-  3  X  52  +  36  X  5  =  —    75  +  180  =  105. 

2.  ar»  — 4a;  +  3  =  0.  Ans.x  =  l    or  3. 

3.  6a:2  _  13a:  =  —  6.  *        Ans.  x  =  ^  or  -, 

4.  a;2  _  52;  _^  4  =  0.  Ans.  x  =zl  or  4. 

6.    3ar5  —  7a;  =  20.  Jt^s.  a;  =  4  or  —  -. 

o 

6.  2r5  —  7a:  +  3  =  0.  Ans.  x  =  3  or  -. 

7.  3arJ  —  53a;  +  34  =  0.  Ans.  a:  =  17  or  | 

o 

8.  a^  -{-  10a;  +  24  =  0.  ^W5.  a;  =  —  4  or  —  6. 

9.  {x  —  l)(x  —  2)  =  6.  Ans.  a;  =  4  or  —  1. 

10.  (3a;  —  5)  (2a;  —  5)  =  (a;  +  3)  (a;  -  1). 

Ans.  a;  =  4  or  -. 
5 

11.  (2a;  —  3)2  =  8a;.  Ans.  ^  =  1  ^^  l- 
1^-    ^  =  ^-^-                          ^....  =  51  or  5. 

13.  V(2a;  +  7)  +  ^{3x  -  18)  =  ^{7x  +  1). 

A71S.  x=z9  or  —  3|. 

14.  aa?*—  ac  =  ex  —  03^.  Ans.  x  =  -^= -— — ! — . 

2{a-\-b) 

15.  a2+Z>2_2Ja;  +  a;2  =  ^. 

n^ 

Ans.  X  =  -J— — ^^{bn  ±  ^/ahr?  +  Ithn?'  —  a^n%. 


COMPLETE    EQUATlOi^S.  229 

16.  3x-^  4-  2x-^  =  1.  Ans.  a;  =  3  or  —  1. 

17.  4o-4i  =  2a;0.  Ans.  x=l  ov  -% 
X  ^      x^  5 

18.  3x-^2Vx=  16.  A71S.  x  =  '7^  ov  4. 
12 


19.     Va;  +  5  =  -—r .  ^7^5.  ic  =  4  or  —  21. 

Vx  +  12 

21.  V^  4-  m  —  a/^+ /^  =  \/2i. 

Ans.  x= —  ±  -  V2m^  +  2nK 

22.  (X  -  c)  (ab)^^  ^^  =  0.  Ans.  ^  =  f  or  ^'. 

{cx)-^                                      *  ^ 

23.  ^         +  1         ^  12a  (g  +  x)~^ 

{a-hx)~^      (a-x)'^  ^ 

J              4:a  Sa 

Ans.  x  =  --  or  -=-. 

5  5 

a:- V^+l        5                                ^               Q  8 

x-^Vx  +  1       11  9 


25.    — :  +  hx-^  =  — ,  •  Ans.  x  =  — =^-^^t 

359.  When  a  complete  equation  of  the  second  degree  is  pro- 
posed for  solution,  instead  of  going  through  the  process  of  com- 
pleting the  square,  we  may  use  the  formula  x=  — ^±y$'+^. 
For  example,  take  the  equation 

—  3a^  +  36a;  =  105. 

Dividing  by  —  3,      a^«  —  12a;  =  —  35. 

In  this  case,  p  =  —12,  and  q  =:  —35;  hence,  by  the  for- 
mula, 

x= ±y  —  35  +  ^^ — j—^  =  7  or  5. 


230  QUADEATIC    EQUATIONS. 

EXAMPLES. 

Solve  the  following  equations  by  using  the  formula 

1.  a;2  — 6ir=7. 

2.  a?  -{-l^  =  95. 

3.  x^—2x  =  ^, 

4.  a?  +10x=  —  9. 
6.  0^2  _  14^ —  120. 

6.  a:2  _^  32a;  =  320. 

7.  0^2  ^  lOOrr  =  1100. 

8.  x'-x  =  \. 

4 


9.    a;2  ^  3^^:  _  19. 

13 
5 


''*i- 

Ans 

.  X 

;=:7  or 

—  1. 

Ans. 

X: 

=  5  or  - 

-19. 

Ans 

.   X 

:  =r  4  or 

-2. 

Ans.  X 

= 

—  1  or 

-9. 

Ans. 

X: 

=  20  or 

-6. 

Ans. 

X  : 

=  8  or  - 

-40. 

Ans.  X 

= 

10  or  - 

110. 

Ans. 

X  : 

=  li  or 

1 
2* 

Ans. 

X  : 

=  3  or  - 

-6i. 

1  Q 

10.    xi  +  ^x  =  14u  Ans.  ic  =  7|  or  —  10. 


11.  2iC  =  4  4-  -.  Ans.  a;  =  3.  or  —  1. 

X 

x^  —  S  1 

12.  X ^— — r  =  2.  Ans.  xz=:2  OT  -. 

x^  -{-  6  2 

a:2  X      m^  —  4a' 


13. 


3w  —  2a       2      4a  —  6m 


^W5.  a;  =  m  —  2a  or  ^m  -\-  a. 

14.  ^_^  =  a.  ^««.  rr  =  5(2±A/^+l). 
x—d      x-\-6  a^ 

15.  mrc^ a;  =  1.  Ans.  x=z—  or =. 

mn  n  m^ 

16.  ^[a^5  +  a(a4.J)]+^da:  =  ^a;(20a4-73). 

Ans.  a;  =  2a  or  2  (a  +  ^). 


COMPLETE    EQUATION'S.  231 

360.  Solving  the  equation  ax^  -^  bx  =  c  in  the  usual  way, 
we  find 

_^b±Vb^  -{-  4:ac 
""-  2a 

To  solve  an  equation  of  the  second  degree  by  means  of  this 
formula,  it  is  only  necessary  to  reduce  it  to  the  form  of  ax^-\-bx=Cf 
and  then  make  the  proper  substitutions.  For  example,  take  the 
equation  —3cc^-{-  36x  =  105.  In  this  example,  a  =  —  3,  ^  =  36, 
and  c  =  105 ;  hence,  by  the  formula, 


_  —  36±  V362+4(— 3)105  _  —  36  J:  Vl296  —  1260 
-^^±^--^^±^  =  6Tl  =  5or7. 


—  6  —6 

Let  the  student  solve  some  of  the  equations  of  Articles  358 
and  359  in  this  way. 

361.  To  complete  the  square  by  the  Hindoo  Method. 
Take  the  equation 

aa?  -{-hx  =  c    .    .    .    (1). 
Multiplying  both  members  by  4«, 

^ah?  +  A:abx  =  ^ac    .    .    ,    (2). 
Adding  l^  to  both  members  of  (2), 

^aho^  +  ^abx  -\- If^  =  4.ac -\- b^    .    .    .     (3). 
Extracting  the  square  root  of  both  members  of  (3), 


2ax  +  h=  ±V4:ac  +  b^    .    .    .     (4); 


whence,  ^^-•h  ±^^ac +  b^    .    .    .    ^^y 

Solve  by  this  method  the  following  equation : 
bx^  —  3x  =  224. 


232  QUADRATIC    EQUATIONS. 

Multiplying  both  members  by  4  x  5,  the  given  equation  be- 
comes 

lOOa^  —  60a;  =  4480. 

Adding  3^  to  both  members  of  this  equation, 

100^:2  _  60a;  +  9  =  4489. 

Extracting  the  square  root, 

10a;  —  3  =  _^  u7 ; 

3  4-  67 
whence,  x  =     ^      =  7  or  —  6^. 

Let  the  student  solve  some  other  equations  in  this  way. 

363.  To  cause  the  term  containing  the  first  powei 
of  the  unknown  quantity  to  disappear. 

If,  in  the  equation 
we  substitute  z  —  ^  for  a;,  we  obtain 


(._|)V,(._|).,, 


that  is,  z2_^ 


^'-■J  =  r 


whence,  zz=±)/q-\-^; 


Solve  by  this  method  the  following  equation : 
a^5  —  11a;  =  —  18. 

Substituting  z  -\-  —  for  x,  this  equation  becomes 


PEOBLEMS.  233 


I 


whence,  ^  =  -7-, 


and  ^"==^2' 

11      7 
=.  =  ^±3  =  9  or  2. 


363.  pmobijEms. 

1.  Find  a  number,  such  that  the  square  of  one- tenth  of  it 
shall  be  equal  to  the  remainder*  obtained  by  subtracting  24  from 
the  number. 

Let  X  =  the  number ; 

then,  by  the  problem, 

whence,  x  =  60  or  40. 

2.  Divide  the  number  10  into  two  parts,  such  that  their  pro- 
duct shall  be  24, 

Let  X  =  one  part ; 

then  will  10  —  a;  =  the  other  part 

Hence,  by  the  problem, 

x{10  —  x)  =  24:; 
whence,  a;  =  4  or  6 ; 

therefore  10  —  a;  =  6  or  4. 

Here,  although  x  may  have  either  of  two  values,  yet  there  is 
only  one  answer  to  the  problem ;  one  part  must  be  4  and  the 
other  6. 

3.  A  person  bought  a  certain  number  of  oxen  for  $400.  If  he 
had  bought  4  more  for  the  same  sum,  each  ox  would  have  cost  $5 
less.    How  many  did  he  buy  ? 


234  QUADRATIC    EQUATIONS. 

Let  X  =  the  number  of  oxen ; 

then  will  —  =  the  cost  of  each  in  dollars. 

X 

If  he  had  bought  4  more  for  the  same  sum,  the  cost  of  each 
would  have  been 


a;  +  4' 


400         400       ^ 

-: — 5; 


a;  4-  4        a; 
whence,  x=16  or  —  20. 

Only  the  positive  value  of  x  is  admissible ;  hence,  the  number 
of  oxen  is  16. 

In  solving  problems  by  algebra,  results  will  sometimes  be  ob- 
tained which  do  not  apply  to  the  question  actually  proposed.  The 
reason  is  that  the  algebraic  language  is  more  general  than  ordi- 
nary language,  and  thus  the  equation,  which  is  a  proper  expression 
of  the  conditions  of  the  problem,  is  also  appUcable  to  other  con- 
ditions. It  is  sometimes  possible,  by  making  suitable  changes  in 
the  enunciation  of  the  original  problem,  to  form  a  new  problem, 
corresponding  to  any  result  which  was  inapplicable  to  the  original 

400 


problem.    If  we  change  the  sign  of  x  in  the  equation 


X-\-4: 


400       ,     .^,                  400         400       ,           400         400       ^ 
5,  it  becomes  -. = 5,  or = f-  5. 

X  4:  —  X  —X  X  —  4:  X 

This  equation  is  the  algebraic  statement  of  the  following  prob- 
lem :  A  person  bought  a  certain  number  of  oxen  for  $400.  If  he 
had  bought  4  less  for  the  same  sum,  each  ox  would  have  cost  $5 
more.    How  many  did  he  buy  ? 

o  1  •      XI.  J.'         400         400       ^ 

Solving  the  equation   j  = 1-  5,    we  find  a;  =  20    or 

X    —    TC  X 

—  16. 

In  this  connection  the  student  should  review  Art.  216, 

4.  Find  two  numbers  whose  difference  is  8  and  whose  product 
is  240.  Ans.  12  and  20. 

6.  Find  two  numbers  whose  difference  is  2a  and  whose  product 
is  h,  Ans.  a  ±  V«^  -|-  h  and    —  a  i  ^/d?  -f  h. 


PROBLEMS.  235 

6.  The  remainder  obtained  by  subtracting  a  certain  number 
from  10  is  equal  to  the  quotient  obtained  by  dividing  25  by  that 
number.    What  is  the  number  ?  Ans,  6, 

7.  Divide  the  number  40  into  two  such  parts  that  their  pro- 
duct shall  be  equal  to  15  times  their  difference. 

Ans.  60  and  —  20,  or  10  and  30. 
The  numbers  60  and  —  20  satisfy  the  problem  in  the  algebraic 
sense,  but  not  in  the  arithmetical  sense. 

8.  Divide  a  into  two  such  parts  that  their  product  shall  be 

equal  to  m  times  their  difference. 

a  —  2m±  Va^  +  ^^^        .    «  +  2m  =f  Va^  +  4m2 
A71S.  ==-^ and ^ 

9.  Di\4de  100  into  two  such  parts  that  the  sum  of  their  square 
roots  shall  be  14.  Ans.  64  and  36. 

10.  Divide  a  into  two  such  parts  that  the  sum  of  their  square 
roots  shall  be  s. 


a  +  V2as^  —  s*      j  «  —  V2as^  —  s* 
Ans,  2 ^^^  2 * 

11.  A  and  B  start  at  the  same  time  from  different  places  and 
travel  toward  each  other.  At  the  end  of  14  hours  they  meet, 
when  it  appears  that  A  has  traveled  10  miles  more  than  B,  and 
that  their  rates  of  travel  are  such  that  B  requires  half  an  hour 
more  than  A  to  travel  20  miles.     Find  B's  rate  of  travel. 

Ans.  5  miles. 

The  negative  result  is  rejected,  because  it  does  not  satisfy  the 
problem  in  its  arithmetical  sense. 

12.  A  and  B  start  at  the  same  time  from  different  places  and 
travel  toward  each  other.  At  the  end  of  m  hours  they  meet, 
when  it  appears  that  A  has  traveled  a  miles  more  than  B,  and 
that  their  rates  of  travel  are  such  that  B  requires  n  hours  more 
than  A  to  travel  b  miles.    Find  B's  rate  of  travel. 

A  a    ^      /  ab    ^     a^ 

Ans.  —^ hi/ h  T— 5' 

2m       ^   m?i      4m2 


236  QUADRATIC    EQUATIONS. 

13.  A  started  from  C  toward  D,  and  traveled  at  the  rate  of  10 
miles  an  hour.  When  he  was  9  miles  from  C,  B  started  from  D 
toward  C,  and  went  every  hour  one-twentieth  of  the  distance  from 
D  to  0.  When  B  had  traveled  as  many  hours  as  he  went  miles  in 
one  hour,  he  met  A.    Find  the  distance  from  C  to  D. 

Ans,  180  miles  or  20  miles. 

14.  A  went  from  C  to  D,  traveling  a  miles  an  hour.  When  he 
was  b  miles  from  C,  B  started  from  D  toward  C,  and  went  every 

hour  -  th  of  the  distance  from  D  to  C.    When  B  had  traveled  as 
n 

many  hours  as  he  went  miles  in  one  hour,  he  met  A.     Find  the 
distance  from  0  to  D.  Tn  —  a       ,  /'/n^^^^     ~~\ 

15.  A  and  B  were  traveling  on  the  same  road,  and  at  the  same 
rate,  from  Columbia  to  St.  Louis.  At  the  50th  mile-stone  from 
St  Louis,  A  overtook  a  flock  of  geese  which  were  traveling  at  the 
rate  of  three  miles  in  two  hours,  and  two  hours  afterward  met  a 
wagon  which  was  moving  at  the  rate  of  nine  miles  in  four  hours. 
B  overtook  the  same  flock  of  geese  at  the  45th  mile-stone,  and 
met  the  same  wagon  40  minutes  before  he  reached  the  31st  mile- 
stone.    Where  was  B  when  A  reached  St.  Louis  ? 

Ans.  25  miles  from  St.  Louis. 


THEORY  OF  QUADRATIC  EQUATIONS  WITH  ONE  UNKNOWN 

QUANTITY. 

364,  Every  equation  of  the  second  degree  containing  only  one 
unknown  quaiitity  has  two  roots,  and  only  two. 

Every  equation  of  the  second  degree  containing  only  one  un- 
known quantity  can  be  reduced  to  the  form  of 

x^  -\-  px=z  q. 

Solving  this  equation,  we  find 


-|±/^+f 


has 

THEOKEMS.                                                    237 

Hence  x 

two  values,  namely:    —  "|  +  y  S'  +  x     and 

Denoting 
we  have 

thef 

irst  of  these  values  by  a;'  and  the  second  by  x"y 

.■=-t+^,+t, 

^--i-/,.?- 

The  equation  ot^  -^  px  =  q  cannot  have  more  than  two  roots. 
If  possible,  let  a,  b,  and  c  be  three  different  roots  of  this  equa- 
tion ;  then  will  these  roots  satisfy  the  equation. 


a^  -\-  pa  =  q    .    .    . 

(1), 

l^+pb  =  q    .    .    . 

.     (2), 

c^  -]-  pc  =  q    .     .     . 

,     (3). 

Subtracting  (2)  from  (1), 

a^  —  ^2^^(^a-b)=0    , 

.    .    . 

Subtracting  (3)  from  (1), 

a2  —  c^^p(^a  —  c)  =  0    . 

.    .    . 

(4). 


(5). 

Dividing  both  members  of  (4)  by  a  —  b,  which  is,  by  hypoth- 
esis, not  zero,  we  obtain 

a-{-b-{-p=0    .    .    .    (6). 

Dividing  both  members  of  (5)  by  a  —  c, 

a  +  c+p  =  0    .    .    .    (7). 

Subtracting  (7)  from  (6), 

b  —  c  =  0; 

whence,  5  =  c ; 

that  is,  two  of  the  supposed  roots  are  equal  to  each  other ;  there- 
fore the  equation  x^  -\- px  =  q  cannot  have  three  different  roots. 


238  QUADRATIC    EQUATIONS. 

365.  The  sum  of  tlie  roots  of  an  equation  of  the  form  of 
Qi?  -\-  px=z  q  is  equal  to  the  coefficient  of  the  second  term  taken 
with  the  contrary  sign. 

Solving  the  equation  a^  +  px=.q,  we  obtain 


^■=-|  +  iVf^  •  •  •  «' 


XX 


and  -„"=_|_yVh|    •    •    •     (2)- 

whence,  by  addition, 

X  -f  x"  =  —p. 

Thus,  the  roots  of  the  equation  a?  —10xz=z  -^1^  areS  and  2, 
and  their  sum  is  10. 

366.  The  product  of  the  roots  of  an  equatio7i  of  the  form  of 
x^  -\-  px  =  q  is  equal  to  the  second  member  taken  with  the  coti- 
trary  sign. 

From  (1)  and  (2)  of  Art.  365  we  obtain,  by  multiphcation, 

(-|  +  i^)(-|-v^)=?-(«+?) 

Thus,  the  roots  of  the  equation  a?—  10x=  —  16  are  8  and  2, 
and  their  product  is  16. 

Cor. — The  independent  term  q  is  divisible  by  each  of  the 
roots. 

367.  Every  equation  of  the  second  degree  containing  onlp  one 
unknown  quantity  can  be  reduced  to  the  form  of 

(x  -  x')  (x  -  x")  =  0. 

Denoting  the  roots  of  the  equation 

a^-\-px  =  q    .    .    .     (1), 

by  a'  and  x",  we  have 

p=-{x'-\-x"\ 

and  q  =  --  x'x" ; 


APPLICATION    OF    THEOREMS.  239 

hence  (1)  becomes 

if2  _  (^x'  +  x")  x=—  x'x"    .    .    .     (2). 

By  transposition  and  factoring,  (2)  becomes 

(x  —  x')  (x  —  x")  =  0    .    .    .     (3). 

Cor. — Hence  a^  -\-  px  —  q  ^  {x  —  x')  (x  —  x") ;  therefore  the 
first  member  of  the  equation  x^  -\-  px  —  q  =  0  is  divisible  by 
X  —  x'  and  by  a;  —  x", 

368.  EXAMPI.E8. 

1.  Find  the  equation  whose  roots  are  2  and  3. 

1st  /SoZw^iow.— Substituting  —(2  +  3)  for ^  (365), and  —2x3 
for  q  (366),  the  general  equation  7?  -\-  px  =z  q  becomes 

x^-hx=  —  Q. 

2d  Solution. — Substituting  2  for  x  and  3  for  x",  (3)  of  Art. 

367  becomes 

(a;_2)(ic-3)  =0; 

that  is,  a:2  _  5;^;  +  6  =  0. 

2.  Resolve  the  first  menaber  of  the  equation  nf  -^  6a;  -f  8  =  0 
into  two  binomial  factors. 

Solving  this  equation,  we  find  a:'  =  —  2,  a;"  ir:  ~  4;  hence 
the  given  equation  may  be  wrtteo  in  the  form 

[z-(-2)][a;-(^4)]=  0(367), 
that  is,  {x  +  2)  (a;  +  4)  =  0. 

3.  Find  the  equation  whose  roots  are  5  and  2. 

Ans.  x^—'ixz=z  -10- 

4.  Find  the  equation  whose  roots  are  3  and  3. 

Ans.  a:2  —  6a;  =  —  9. 

44 
6.  Find  the  equation  whose  roots  are  10  and  — —, 

o 

Ans.  3?  ■\-  -3--'^*  =  -g-- 


240  QUADRATIC    EQUATIONS. 

n   -^   :.  1.^.  i_-         1-  X  13+V85      ^  13  — \/85 

6.  Find  the  equabon  whose  roots  are and  — 

^       ^      13  3 

Ans.  a^  — ;z-x=  —  j^ 

7  7 

7.  YmA  the  equation  whose  roots  are  5  +  V—  1  and  5— V—  1. 

Ans.  2:2  —  10a;  =  —  26. 

8.  Resolve  the  first  member  of  the  equation  Sa;^— 10a;— 25=0 

into  three  factors,  .       _ ,        r\  /     .   5\       „ 

Ans,  d{x  —  5)lx-\--j  =  0. 

9.  Resolve  the  first  member  of  the  equation  3^-\-'ildx-\-780=z0 
into  two  binomial  factors.  Ans.  {x  -f  60)  {x  +  13)  =  0. 

10.  Resolve  the  first  member  of  the  equation   2a;2+  a:  —  6  =  0 

into  three  factors.  .        « /     .   «x  /        3\ 

Ans.  2  (a;  +  2)  (a;  —  -I  =  0. 

11.  Resolve  the  first  member  of  the  equation  a;2— 88a; + 1612 = 0 
into  two  binomial  factors.  Ans.  {x  —  62)  (x  —  26)  ==  0. 

12.  Resolve  the  first  member  of  the  equation  x^  +  a^  =  0  into 
two  binomial  factors.      Ans.  (x  —  a  V^^) (a;  +  «  a/^^)  =  0. 


DISCUSSION  OF  THE  EQUATION  a;2  -f  ^a;  =  q. 

369.  The  Discussion  of  an  equation  consists  in  making 
every  possible  supposition  with  regard  to  the  arbitrary  quantities 
contained  in  it,  and  interpreting  the  results. 

The  arbitrary  quantities  in  the  equation  ^  ^  px^q  are  ^ 
and  q, 

370.  We  shall  first  make  every  possible  supposition  in  rela- 
tion to  the  signs  of  'p  and  q. 

Suppose, /rs^,  that  j9  and  q  are  positive;  second,  thatjt?  is  neg- 
ative and  q  positive ;  tliird,  that  p  is  positive  and  $-  negative ; 
fourth,  that  p  and  q  are  negative.     We  shall  thus  have 


DISCUSSION. 

The  Four  Forms 

x^-\-px=z       q    .    . 

•  m, 

Q^—px—       q    .    . 

(2), 

a?  -\-  px=:  —  q    .    . 

(3), 

a?  —  px=  —  q    .    , 

(4). 

241 


371.  In  the  first  form  one  root  is  positive,  the  other  negative, 
and  the  negative  root  is  numerically  the  greater. 

Since  q  is  positive,  the  product  of  the  roots  is  negative  (366) ; 
•  hence  one  root  is  positive  and  the  other  negative.  Again,  since 
p  is  positive,  the  sum  of  the  roots  is  negative  (365) ;  hence  the 
negative  root  is  numerically  the  greater. 

Illustration. — The  roots  of  the  equation  a?  -\-  x=iQ  are  2 
and  —  3. 

372.  In  the  second  form  one  root  is  positive,  the  other  nega- 
tive, and  the  positive  root  is  numerically  the  greater. 

Since  q  is  positive,  the  product  of  the  roots  is  negative ;  hence 
one  root  is  positive  and  the  other  negative.  Again,  since  p  is 
negative,  the  sum  of  the  roots  is  positive ;  hence  the  positive  root 
is  numerically  the  greater. 

Illustration. — The  roots  of  the  equation  a?  —  x=z  210  are  15 
and  — - 14. 

373.  In  the  third  form  both  roots  are  negative. 

Since  q  is  negative,  the  product  of  the  roots  is  positive ;  hence 
they  have  like  signs ;  and  since  p  is  positive,  the  sum  of  the  roots 
is  negative ;  hence  both  roots  are  negative. 

Illustration. — The  roots  of  the  equation  x^  -^  ^x  =i  —  12  are 
—  4  and   —3. 

374.  In  the  fourth  form  both  roots  are  positive. 

Since  q  and  p  are  negative,  the  product  and  the  sum  of  the 
roots  are  positive ;  hence  both  roots  are  positive. 

Illustration. — The  roots  of  the  equation  a?  —  "^x^^  —  1%  are 
4  and  3.  .  " 

16 


243  QUADRATIC    EQUATIOJN^S. 

375,  For  convenient  reference,  the  four  fonns  and  their  cor- 
responding roots  are  here  given. 


^  -\-  px^q    .    .    .    (1) ;  whence 


Q?-^px=:q    .    .    .    (2);  whence 


-'•=i+i/?+f 


=^"=i-i/.+^- 


oi^j^ px:^^q    .    .    .     (3);  whence 


ci^^px^^q    .    .    .     (4);  whence 


^-hV^ 


376.  Unequal  Moots,— The  roots  of  an  equation  of  the 
first  or  of  the  second  form  are  unequal,  whatever  the  relative 
values  of  j9  and  q  may  be  (371-373). 

The  roots  of  an  equation  of  the  third  or  of  the  fourth  form  are 

unequal  if  -^  is  greater  or  less  than  q. 

377.  Equal  Hoots, — The  roots  of  an  equation  of  the 
third  or  of  the  fourth  form  are  equal  if  ^—  is  equal  to  q. 

Illustration,— ^oWing  the  equation  a;^  -|-  6a;  =  —  9,  we  find 
a;'  =  —  3  and  x"  =  —  3.  Solving  the  equation  a;^  —  Ga;  =  —  9, 
we  find  x'  =  3   and  x"  =  3. 

378.  Heal  JSoo^«.— The  roots  of  an  equation  of  the  first 


DISCUSSION.  243 

or  of  the  second  form  are  real,  whatever  the  relative  values  of  p 

«2  .         .  . 

and  q  may  be,  for  in  these  forms  -r  ■\-  q   '^^  positive. 

The  roots  of  an  equation  of  the  third  or  of  the  fourth  form  are 

real  if  -r  is  not  less  than  a, 
4 

Remabk. — The  quantities  jP  and  q  are  liere  supposed  to  be  real. 

379.  Imaginary  Moots, — The  roots  of  an  equation  of 

the  third  or  of  the  fourth  form  are  imaginary  if  —■  is  less  than  q'y 

for  the  radical  part  of  each  of  the  roots,  in  this  case,  is  the  square 
root  of  a  negative  quantity. 

Illustration. — The  roots  of  the  equation  a;^  +  6a;  =  —  10  are 
__  3  ^  V-- i  and  —  3  —  V—  1 ;  and  the  roots  of  the  equation 
0^2  _  6a;  =  —  10   are   3  +  V^^  and  3  —  V~-^. 

380.  Imaginary  roots  indicate  incompatible  conditions. 

The  demonstration  depends  upon  the  following 

Lemma. — 77ie  greatest  product  wJiich  can  be  obtained  by  sep- 
arating a  given  number  into  ttvo  parts  and  multiplying  one  by 
the  other  is  the  square  of  half  that  number. 

Let  p  be  the  given  number,  and  d  the  difference  of  the  parts 
into  which  it  is  separated. 

7)       d 
Then         -^  +  -  =  the  greater  part, 

10      d 
and  -g  — -  -  =  the  less  part  (213,  4). 

Denoting  the  product  of  the  parts  by  P,  we  have 

4       4' 
Now,  since  jO  is  a  given  number,  it  is  evident  that  P  will  in- 


244  QUADKATIC    EQUATIONS. 

crease  as  d  diminishes,  and  will  be  the  greatest  possible  when 
d  =  0\  that  is,  P,  when  greatest,  is  equal  to   (^  . 

Illustration.        8  =  1  +  7;  7x1=    7.      . 

8  =  2  +  6;  6  X  2  =  12. 

8  =  3  +  5;  5x3  =  15. 

8  =  4  +  4;  4x4=16. 

In  the  first  form  the  sum  of  the  roots  is  —  ^  (365),  and  their 
product  is   —  q  (366) ;   hence  (Lem.),  in  this  form,  —  q  can- 

not  be  greater  than  -^. 

In  the  second  form  the  sum  of  the  roots  is  p,  and  their  pro- 
duct is  —  5' ;  hence,  in  this  form,  —q  cannot  be  greater  than  '-j-- 

In  the  third  form  the  sum  of  the  roots  is  —  jo,  and  their  pro- 
duct  is  q;  hence,  in  this  form,  q  cannot  be  greater  than  ^. 

In  the  fourth  form  the  sum  of  the  roots  is  p,  and  their  product 

is  q ;  hence,  in  this  form,  q  cannot  be  greater  than  ■^. 

In  the  first  and  second  forms  q  is  positive;  hence  an  equation 

p2 
in  which  —  q  is  greater  than  ■'—  can  never  occur  in  either  of 

these  forms ;  for  ^—  being  positive  is  greater  than  any  negative 

quantity.    But  in  the  third  and  fourth  forms  an  equation  may 

«2 
occur  in  which    +  5'  is  greater  than  •'—.    Thus,  in  the  equations 

x^  +  6x=  —10  and  a?^—(},x=—  10,  the  independent  term, 
taken  with  the  contrary  sign,  is  greater  than  the  square  of  half 
the  coejQ&cient  of  the  second  term ;  therefore  the  roots  are  imag- 
inary. 

Hence,  if- any  problem  furnishes  an  equation  of  the  third  or  of 
the  fourth  form,  in  which  the  independent  term,  taken  with  the 
contrary  sign,  is  greater  than  the  square  of  half  the  coefficient  of 


PEOBLEM    OF    THE    LIGHTS.  245 

the  second  term,  we  infer  that  the  problem  contains  incompatihle 
conditions. 

For  example,  let  it  be  required  to  find  two  numbers  whose  sum 
shall  be  6  and  product  10. 

Let  X  =  one  of  the  numbers ; 

then  will  6  —  a;  =  the  other. 

By  the  second  condition  of  the  problem, 
x((o  —  x)=ilO', 
that  is,  Qx  —  x^  =  10, 

or,  by  changing  signs,    t?  —  6.r  =  —  10 ; 

whence,  a;  =  3  ±  V—  1. 

The  imaginary  roots  indicate  that  there  are  no  real  numbers 
whose  sum  is  6  and  product  10.  The  greatest  product  which  can 
be  formed  by  separating  6  into  two  parts  and  multiplying  one  by 
the  other  is  9. 

PROBLEM  OP  THE  LIGHTS. 

381.  To  find,  on  the  straight  line  joining  tivo  lights,  the 
points  which  are  equally  illuminated  hy  those  lights. 

! I [ I I 

P''  A  P  B  P' 

Let  A  and  B  be  the  two  lights.  Denote  the  intensity  of  the 
light  A  at  a  unit's  distance  by  a,  the  intensity  of  the  light  B  at  a 
unit's  distance  by  h,  and  the  distance  between  the  lights  by  d. 

Let  P  be  a  point  equally  illuminated  by  the  two  lights,  and  let 
ic  =  AP;  then  will  d  —  x  =  BP. 

One  of  the  laws  of  light  is,  that  the  intensity  of  a  light  at  any 
distance  as  x,  is  equal  to  the  quotient  obtained  by  dividing  its  in- 
tensity at  a  unit's  distance  by  x^\  hence 

-^  =  the  intensity  of  the  light  A  at  P, 
and  -7-^ i-g  =  the  intensity  of  the  light  B  at  P. 

yCL  —  Xj 


246  QUADRATIC    EQUATIONS. 

But  P  is  to  be  equally  illuminated  by  the  two  lights; 
±-        ^  ,    .    .     (1). 

Clearing  this  equation  of  fractions, 

a{d-x)^  =  bx^    .    .     .     (2). 
Extracting  the  square  root  of  both  members  of  (2), 
(d-x)Va=  ±xVb    .    .    .     (3) ; 


Wa  ±  VbJ 


whence, 

Separating  the  values  of  x, 

Wa  +  Vb/ 

and  a^''=d(-J^-\ 

\Va  -  Vb' 

From  the  nature  of  the  problem,  a  and  b  are  positive ;  hence 
the  values  of  x  are  real ;  therefore  there  are  two  points  of  equal 
illumination,  and  only  two,  on  the  line  of  the  lights. 

Six  different  suppositious  can  be  made  upon  the  arbitrary 
quantities  a,  b,  and  d^  namely : 

1.  ayb    and    dyO.  4:,    ay  b    and    d  =  0. 

2.  a  =  b    and    ^  >  0.  5.    a  =  b    and    d  =  0. 

3.  a  <.b    and    c? >  0.  6.    a  <b    and    d  =  0. 

1.     ay  b    and    c? >  0. 

In  this  case  — = —  is  a  proper  fraction ;  that  is,  it  is  less 

V«  +  Vb 
than  1 ;  and  since  the  denominator  is  less  than  twice  the  numera- 
tor, the  fraction  is  greater  than  ^. 


PROBLEM    OF    THE    LIGHTS.  247 


<  d  and  >  -d. 


The  point  P  is  therefore  between  the  two  Hghts  and  nearer  the 
weaker  one. 

The  fraction  ^^    -  >  1 ; 


ivBvh' 


V«  —  vh) 

The  second  point  of  equal  illumination  is,  therefore,  at  some 
point  P'  on  the  right  of  B. 

2.    a=zb    and    e^  >  0. 

In  this  case  x'  =  -  and  x"  =  -——  =  oo  (222,  1) ;  that  is, 

the  first  point  of  equal  illumination  is  at  the  middle  point  of  AB, 
and  the  second  is  at  an  infinite  distance  to  the  right  of  A.  The 
symbol  oo  indicates  impossihility ;  that  is,  it  shows  that  there  is 
no  second  point  of  equal  illumination. 


3.    a<,b    and    d>0. 

In  this  case  —-= —  <  ^  and  -— —  is  negative ; 

Va  +  V  6      ^  Va  —  yb 

x' <C-d    and    a;"  <  0. 

The  first  point  of  equal  illumination  is,  therefore,  between  the 
lights  and  nearer  to  A,  and  the  second  point  is  at  some  point  P" 
on  the  left  of  A. 

4.    a  >  5    and    d  =  0. 

In  this  case  x'  =  0  and  x"  =  0. 

How  are  these  results  to  be  interpreted  ?    They  seem  to  indi- 


248  QUADKATIC    EXPEESSIONS. 

cate  that  the  point  at  which  the  lights  are  placed  is  equally  illu- 
minated by  them ;  but  this  is  not  true,  as  we  shall  see  by  consid- 
ering equation  (1).    Under  the  hypothesis  that  d=zO,  (1)  becomes 


^2       a;2  • 

But  this  is  not  an  equation  in  fact,  for  a'>l),  and  the  de- 
nominators are  equal.    It  would  not  be  an  equation  if  a;  =  0,  for 

then  -^  and  -^  become  unequal  infinities. 

There  is,  then,  in  this  case,  no  eqiiation,  and  hence  no  point 
of  equal  illumination. 

5.  az=l)    and    J  =  0. 

In  this  case  a;'  =  0  and  x"  =  -. 

The  first  value  of  x  indicates  that  the  point  at  which  the  two 
equal  lights  are  placed  is  equally  illuminated  by  them,  and  the 
second  value  of  x  indicates  that  a7iy  point  on  the  line  of  the 
lights  is  equally  illuminated  by  them  (322,  4).  As  the  lights 
are  now  at  the  same  point,  the  line  of  the  lights  may  be  drawn  in 
any  direction  in  space ;  hence,  in  this  case,  any  point  in  space 
will  be  equally  illuminated  by  the  two  lights. 

6.  a  <ib    and    d  =  0. 

In  this  case  x'  =  0  and  x"  =  0. 

But  under  this  hypothesis,  as  in  case  4,  equation  (1)  becomes 
impossible;  hence  no  point  in  space  is  equally  illuminated  by  the 
two  lights. 

QUADRATIC  EXPRESSIONS. 

383.  A  Quadratic  Expression  is  a  trinomial  of  the 

form  of 

ax^  -\-  bx  +  c; 

in  which  a,  b,  and  c  represent  given  numbers,  positive  or  negative, 
and  in  which  x  may  have  any  value.    Thus, 


QUADRATIC    EXPRESSIONS.  249 

ox^  —  dx  +  Q 

is  a  quadratic  expression. 

A  quadratic  expression  is  sometimes  called  a  Trinomial  of  the 
Second  Degree. 

383.  Distinction  between  a  Quadratic  Bqiia^ 
tion  and  a  Quadratic  Expression. — In  the  quadratic 
equation  ax^  -{-  bx  -\-  c  =  0,  x  has  one  of  two  definite  values 
(364) ;  but  in  the  quadratic  expression  ax^  -\-  hx  -\-  c,  x  may 
have  any  value. 

384.  To  resolve  the  quadratic  expression  ax^-^hx-\-c 
into  its  factors. 


Assume 

:+c= 

/ 
-a  X 
\ 

aa? -]- hx  +  c  =  0 ) 

whence, 

%a 

^,     -b-^/b^-^ac. 

/^-               2« 

.-.      ax^-\-hx 

^b-^^/b^-4.ac\l       -b- 
RULE. 

-V0^—4:ac 
2a 

)• 


Assume  the  given  quadratic  expression  to  be  equal  to  zero,  and 
resolve  the  first  member  of  the  equation  thus  obtained  into  its  fac- 
tors (367). 

EXAMPLES.  ' 

Eesolve  each  of  the  following  expressions  into  its  prime  factors  : 

1.  x^  +  2X'- 120.  6.  x^-^9x  —  90. 

2.  xi  —  9x-{-  14.  7.  a:2  +  12:^;  _  45. 

3.  a;2  +  8a;  +  16.  8.  18a:2  _  9a;  —  2. 

4.  Sx^  —  2x  —  3.  9.  8a;2  —  Qx  +  l, 

5.  Qx^  +  x  —  \,  10.  o^—{2a  —  c)x^  2aC' 


250 


QUADRATIC    EQUATIONS. 


385. 


SYNOPSIS    FOR    REVIEW. 


Eh 

o 

}25 


Equations  op  the  Second  Degree  with  only  one  Unknown 
Quantity. 

General  form  of  Equation  op  Second  Degree  with  only 
one  Unknown  Quantity. 

Quadratic  Equations. 

Complete  And  Incomplete  Equations. 

Solution  of  Incomplete  Equation  and  Rule.  Cor. 

Solution  op  Complete  Equation  and  Rule. 


Application  op  the  fobmxtl^ 


Eh  p5 

o 
o 

M 
CQ 


I 

o* 


2a 


Hindoo  method  op  .  completing  the  square 


Method  op  causing  the  term  containing  the  first  power 
of  the  unknown  quantity  to  disappear. 


Propositions  relating   to 
Quadratic  Equations. 


Discussion  op  the  equation 


Problem  op  the  Lights. 
Quadratic  Expressions  . 


Number  of  roots. 
Sum  of  roots. 
Product  of  roots.    Cor. 

Signs  of  the  arbitrary  quantities. 
The  four  form^^  and  correspond- 
ing roots. 
-^    Unequal  roots. 
Equal  roots. 
Real  roots. 
<  Imaginary  roots. 

"  Difference  between  quadratic  ex- 
pression and  quadratic  equa- 
tion. 

^  To  resolve  into  factors. 


CHAPTEE   XIV. 
HIGHER    EQUATIOlsrS 

WITH  ONE  UNKNOWN  QUANTITY. 


386.  There  are  many  equations  which,  though  not  really  of  the 
second  degree,  may  be  solved  by  processes  similar  to  those  given  in 
the  preceding  chapter.  To  this  class  belong :  1st.  All  equations 
of  the  higher  degrees  which  contain  only  one  power  of  the  un- 
known quantity ;  2d.  All  equations  of  the  higher  degrees  which 
contain  two,  and  only  two,  powers  of  the  unknown  quantity,  and 
in  which  the  exponent  of  one  of  these  powers  is  double  that  of  the 
other.    Such  equations  may  be  reduced  to  one  or  the  other  of  the 

forms 

ax"^  =  c    .    »    ,     (1), 

ax^  ■^hx''  =  c    .    .     .     (2). 

Equation  (1)  is  called  a  pure  equation  of  the  n^^  degree. 
Equation  (2)  is  said  to  have  the  form  of  a  complete  equation  of 
the  second  degree. 

EXAMPLES. 

1.  Solve  the  equation   ax"'  =  c. 

Dividing  both  members  by  a,  and  extracting  the  n^^  root  of 
both  members  of  the  resulting  equation,  we  find 


^  a 

2.  Solve  the  equation  aa?""  4-  i^^  =  c. 

b  c 

By  division,  a^"  -f  -a;'*  =  -  ; 


252  HIGHER    EQUATIONS. 

by  completing  the  square, 

by  extracting  the  square  root, 

«  .    *  _    ,   V^ac  +  b^ 
^   '^2a^^         2a        ' 


by  transposition,       x^  = ^ : 

by  extracting  the  w^  root. 


2a 


3.  Solve  the  equation  a:^  —  9a^  =  —  20. 
Completing  the  square, 

whence,  a^  — ^=±^; 

and  X  =  ±  Vs  or  ±  2. 


4.  Solve  the  equation  x^  —  '7x-\-  ^/x^—  7a;  -f  18  =  24. 
Adding  18  to  both  members, 


ic2  _  7^  _^  18  +  V^--"75+18  =  42. 


Assuming  ^7?  —  7a;  -j-  18  =  ?/,  this  equation  becomes 
2^2  +  y  =  42; 
whence,  ^  =  6  or  —  7. 

We  have  now  the  two  equations 


Va;2  _  7a;  ^  18  =  6, 


Va:2-?a;  +  18=— 7, 
from  the  first  of  which  we  find  a;  =  9  or  —  2,  and  from  the  sec- 
ond, a;  =  i(7±  VT73). 


OifE    UNKNOWN    QUANTITY.  253 

5.  Solve  the  equation 

x-i  -i-  Qx^  +  80a;2  +.213a;  —  2128  =  0     .    .     .     (1). 

We  seek  to  transform  (1)  into  another  equation  such  that  its 
first  three  terms  shall  be  the  square  of  a  binomial,  and  the  remain- 
ing terms  shall  contain  the  first  power  of  that  binomial.  We  see 
that  x^  -\-  Qa^  contains  two  terms  of  the  square  of  x^  +  3x,  and 
that  we  only  need  to  add  9x^  to  it  to  complete  the  square. 
Separating  the  term  SOa;^  into  the  two  parts,  9x^  and  71a:^,  (1)  may 
be  written  thus, 

xi^Gx^-{-  9x^  +  710^  -I-  213a;  =  2128, 
or,  (x^  +  3rc)2  +  71  (x^  +  3x)  =  2128    .    .    .    (2). 

Assuming  x^  -\-  3x  =  y,   (2)  becomes 

2/2  +  71^  =  2128  .     .    .     (3); 


_  71  -j-  V18553 
whence,  y  = ^ , 

We  have  now  the  two  equations 


which  are  easily  solved. 

In  the  answers  to  some  of  the  following  examples  some  of  the 
roots  are  omitted. 

6.  Solve   3^:3  +  42a;*  =  3321.        Ans.  a;  =  9   or   (—41)"^. 

7.  Solve  a;io  +  31a;5  _  32.  Ans.  x=zl  or   —  2. 
a  Solve  x^  —  363^  +  216  =  0.  Ans.  a;  =  2  or  3. 

i       1 
9.  Solve   a;«  —  a;»  + 2  =  0.  Ans.  a;  =  2"  or   (—1)". 

i  1 

10.  Solve  a^  —  13a;2n  _  14         ^ns.  x  =  142«  or  (—  1)^. 


254  HIGHEE    EQUATIONS. 

11.  Solve  xi  +  ~=  3}.  Ans.  a;  =  8  or  ~. 

12.  Solve  x^  —  Ux^  +  40  =  0.     Ans.  x=  ±2ot  ± VTO. 

( ^     -'A  1 

13.  Solve  2\x^  -{-  X  7  =  5.  Ans.  a:  =  2"  or  — . 

14.  Solve  ^J  +  (^)  =  n  (n  -  1). 

Ans.x=±/^^    or    ±  Z^:^. 


15.  Solve  a?5  +  5a:  4-  4  =  5  V^+  5a;  +  28. 


5    .  1 


Ans.  X  =  4c  or   —  9,  or   —  -  ±W—  51. 

<0  M 


387.  PROBLEMS. 

1.  A  vintner  draws  a  certaki  quantity  of  wine  out  of  a  fall 
cask  that  holds  256  gallons,  and  then,  filling  the  cask  with  water, 
draws  out  the  same  quantity  of  liquor  as  before,  and  so  on  for  four 
draughts,  when  there  were  only  81  gallons  of  wine  left.  Supposing 
the  water  and  wine  to  become  thoroughly  mixed  every  time  the 
cask  is  filled,  how  much  wine  did  he  draw  off  the  first  time  ? 

Ans.  64  gallons. 

2.  A  number  a  is  diminished  by  the  x^^  part  of  itself;  the  re- 
mainder thus  obtained  is  diminished  by  the  x^^  part  of  itself;  and 
so  on  to  the  fourth  remainder,  which  is  equal  to  h.    Find  the 

value  of  x.  \/^ 

Ans.  X 


"s/a  —  ^/h 


3.  Find  two  numbers  whose  sum  is  8,  and  the  sum  of  whose 
fourth  powers  is  706.  Ans.  3  and  5. 

4.  Find  two  numbers  whose  sum  is  2a,  and  the  sum  of  whose 
fourth  powers  is  21. 

AnS'  a  +  yVSa^  +  b  —  da^  and  a  —  yVSa^-hb  —  3a^. 


OHAPTEE    XV. 
SIMULTANEOUS    EQUATIOI^S, 


DEFINITIONS. 


388.  The  particular  case  in  which  all  the  equations  of  a 
group  are  of  the  first  degree  has  been  considered  in  Chapter  VII. 
We  propose,  in  the  present  chapter,  to  consider  groups,  each  of 
which  contains  at  least  one  equation  of  a  higher  degree  than  the 
first. 

A  group  containing  only  two  equations  is  sometimes  called  a 
JPair, 

389.  When  the  members  of  an  equation  are  entire  and  ra- 
tmial  with  reference  to  its  unknown  quantities,  the  Def/ree  of 
the  equation  is  the  sum  of  the  exponents  of  the  unknown  quanti- 
ties in  the  term  where  this  sum  is  the  greatest.     Thus, 

4:xy  —  3a;  =  2  —  5?/ 
is  an  equation  of  the  second  degree. 

390.  If  an  equation  of  the  second  degree  involving  only  two 
unknown  quantities,  x  and  y,  contains  all  the  kinds  of  terms  of 
which  it  is  susceptible,  it  can  be  reduced  to  the  form  of 

As^  4-  Bxf/  +  C«/2  4-  D^'  +  Ey  +  F  =  0. 

PAIRS  OF  EQUATIONS  ONE  OF  WHICH   IS  OF  THE  FIRST  AND 
THE  OTHER  OF  THE  SECOND  DEGREE. 

391.  EXAMPLES. 

1.  Solve  the  equations 

x-}-y  =  e     .     .    .     (1), 
a^  +  dxy-i-i/zziU    .    .    .    (2). 


256  SIMULTANEOUS    EQUATIOXS. 

From  (1),  y  =  6  —  rr. 

Substituting  this  value  for  y  in  (3), 

a«  ^.  3a;  (6  —  a;)  +  (6  -  xf  =  44; 
■whence,  a;  =  4  or  2. 

Substituting  these  values  for  x  in  the  equation 
y  =  6  —  rr, 
we  find  y  =  2  or  4. 

2.  Solve  the  equations 

ax-\-hy  =  c    .    .    .    (1), 
Axi  -\-  Bxy  +  Cy^  -{■  Dx  +  Ey  -^¥  =  0    ,    .    .    (2). 

From(l),  yz=^^^^. 

Substituting  this  value  for  y  in  (2), 

A:^  +  B.{^)  +  C(i^)\m  +  -E(^)  +  -F  =  0. 

Clearing  this  equation  of  fractions  and  collecting  similar  terms, 

(A&2  _  Bab  +  Ca2)  a;^  +  (Bhc  -  2Cac  +  D^  -  Eab)  ir  +  Cc^  + 
Mc  +  Fb^  =  0    .     .     .     (3). 

Representing  each  of  the  coefficients  in  this  equation  by  a 
single  letter,  it  may  be  written  thus : 

mx^  -{-  nx  -\-p  =  0    .     .     .     (4). 

This  equation  will  furnish  two  values  for  x ;  then  substituting 

C  —  CiX 

these  in  the  equation  y  =  — 7 — ,  we  shall  have  the  correspond- 
ing values  of  y. 

3.  Solve  the  equations 

x-{-y  =  a    .    .    .    (1), 
xy  =  b^  .    .    .  '  (2). 


FIEST    AND    SECOND    DEGREE.  257 


From  (2),  y  =  ^. 

Substituting  this  value  for  y  in  (1), 


^  +  -  =  a    .    .     .     (3); 


whence,  x  = . 

Substituting  these  values  for  x  in  (1),  we  find 


y  = ^ 

Another  Solution. — The  values  of  x  and  y  are  the  roots  of  thp 
equation 

z2-az=-b^    (365-366); 

^  2 

4.  Solve  the  equations 

x  —  yz=a    .    .    .     (1), 
xy  =  l^  .    .    .     (2). 
Eliminating  ?/, 

x-^  =  a    .    .    .    (3); 


whence,  x  = ^ . 

The  corresponding  values  of  y  are 


2 

Another  Solution, — Put  yz=^v;  then  (1)  and  (2)  become 

X  -{-  V  =  a, 
XV  =  —l?^* 


258  SIMULTANEOUS    EQUATIONS. 

Hence  the  values  of  x  and  v  are  the  roots  of  the  equation 


...           ^or.  =  ''±^J  +  * 

2 

'       ^~                2 

5.  Solve  the  equations 

x^y  =  a    .    .    .     (1),      ' 

0,^  +  ^2  =  ^^   .    .    .     (2). 

Squaring  both  members  of  (1), 

7?^%xy^f^a^    .    .    .    (3). 

Subtracting  (2)  from  (3), 

2x1/ z=z  a^  -  b^    .    .    .     (4). 

Subtracting  (4)  from  (2), 

a^^2xy-\-y^=z2b'-a^    .     . 

■  (5); 

■    (6). 

whence,             ^  —  y  =  ±  V2li^  —a^    .    . 

Combining  (1)  and  (6),  we  find 

a  ±  V2b^  —  a2 

^-             2 

a  T  V2^  —  a^ 
^~             2 

Solve  the  following  pairs  of  equations : 

(a?-2y2  =  71)                                 ix=      67 
^-    (     x  +  y  =  20\                       ^^^•-i2/=_47 

or  13, 

or     7. 

(2a:«  +  a:2/-52^2,,,20)                       ^ 
^'    l2x-3y             =    if             ^''^• 

a;  =  5   or 
y  =  3  01 

37 

4' 

13 

9.  " 

\x  +  y=    100) 
^*    (        a:?/ =  2400) 

(x-\-  y  =  4:\ 

10.  ]^+  ^=  n. 

11.  \^-y=^H. 


FIRST    AND    SECOND    DEGREE.  259 

rc  =  60   or  40, 


X  ^y 


15. 


16. 


17. 


=  1 


Ans, 


a-\-x      b-^y 


i  +  ^i-    8 


iC' 


X-^  ^  i/-2 


34 


x-^      y-^ 

J_     A  —A 

y-i  +  a;-i  ~  ?/-!  "^ 


(?/  =  40   or   60. 

2, 


^?Z5. 


1^  =  2. 


-4/i5.  . 


A71S. 


1^  A 

y  =  y  or  3. 

7  or        5, 
-5   or   -7. 


Ans, 


ic  =  5  or  2, 
3   or  6. 


r  = 
U  = 


^W5.  < 


a:  =  ^  or    —  1, 


2/  =  5  or   -1. 


Ans. 


X  = 


y  = 


^^^'{1  = 


X  =za  or  bf 
b  or  a. 


Ans,  \ 


X  =  6   or  3, 
3   or  5. 


Ans. 


x  =  2  or   —  46, 


cx  = 

\y  = 


3   or 


15. 


260  SIMULTANEOUS    EQUATIONS. 

3 


x  —  y  =  % 
18.    -!        ,  ,       16  J- .  Ans.  i 


"  a;  =  5   or        ^ , 

=  3   or   -5 
4 


19.  ^%+y)-+3(--^)-'-yf.  ^„.J 


o  15 

^~     or      — , 

3 

y=l  or  -— . 


PARTICULAR     SYSTEMS. 

393.  If  one  of  the  given  equations  is  of  a  higher  degree  than 
the  second,  or  if  hoth  are  of  a  higher  degree  than  the  first,  the 
elimination  of  one  of  the  unknown  quantities  usually  leads  to  an 
equation  of  a  higher  degree  than  the  second. 

Illustrations. — If  we  eliminate  y  from  the  equations 

j  ax-\-hy  =  c  ) 

Ka^-^x^y  ^xy  -\-y  =  d) 

we  obtain 

{h  —  a)7?-{-(c  —  a)x^+{c  —  a)x  =  hd  —  c\ 

and  if  we  eliminate  y  from  the  equations 


(         0^^  +  2/2  =  13) 
\x-{-xy  '\-  y  =  \l) 


xy  '\-  y 
we  obtain 

a:*  +  2a:3  _  n^  _  482:  =  —  108. 

Since  we  have  thus  far  had  no  general  method  for  the  solution 
of  equations  of  a  higher  degree  than  the  second,  it  follows  that 
we  cannot  now  give  a  general  rule  for  the  solution  of  pairs  of 
simultaneous  equations  where  one  of  them  is  of  a  higher  degree 
than  the  second,  or  where  both  are  of  a  higher  degree  than  the 
first.  We,  however,  frequently  meet  with  pairs  Qf  ^this*  kind  that 
can  be  solved  by  the  aid  of  special  artifices. 


PARTICULAR    SYSTEMS.  261 

EXAMl^LES, 

1.  Solve  the  equations 

x^^f=z^b    .    .    .    (1), 
xy  =  l%    .    .    .    (2). 
Multiplying  (2)  by  2  and  adding  the  result  to  (1), 

whence,  a;  +  2/  =  ±  7    •    •    •     (3). 

Subtracting  the  equation   2xy  =  24  from  (1), 
{x-yY  =  l', 
whence,  s:  —  y  =  ±1    .    .    .     (4). 

We  have  now  four  groups  to  consider,  namely : 

U-y  =  lP         ,  }x-y=-lS' 

\x_y=      Ij'  ix  —  y=—l) 

Solving  these  four  groups,  we  obtain 


iy=±4:)  i?/=±35 


2.  Solve  the  equations 

ic2  +  2/2=:a2    .     .    .     (1), 
xy  =  b^    .    .    .     (2). 
Multiplying  (2)  by  2  and  adding  the  result  to  (1), 
.(a;  +  y)2  =  «2  +  252; 

whence,  x  +  y=  ±  's/o?  +  W    .    .    .    (3). 

Subtracting  the  equation  "Hxy  —  W  from  (1), 
(.r-.2/)'^=:a2-2&2; 


whence,  ^  —  y^±  V«*  —  22^    .    .    .    (4). 


262  SIMULTANEOUS    EQUATIONS. 

Adding  (3)  and  (4), 


2x=  ±  VoM-^  ±  Va^^^W', 


whence,  a;  =  ±  i  a/^T^  ±  i  Va^  -  2^>2. 

Subtracting  (4)  from  (3), 


2t/=  ±  V«*  +  5^^  =F  V«2  — 2^; 


whence,  2^  =  ±  i  V^  +  2^  =F  i  Va2  —  2^. 

3.  Solve  the  equations 

x^^f  =  a^    .    .    .    (1), 
xy  =  lfi    .    .    .     (2). 
Adding  four  times  the  square  of  (2)  to  the  square  of  (1), 
X^  ■}- 2a^f -^  y*  =  a* -{- ^b*    .    .    .     (3) ; 


whence,  3^  +  y^=  ±  Va*  +  U*    .    .    .    (4). 

Adding  (1)  to  (4), 

2xi=za^±  V«*  +  4^; 


whence,  x=±  ^cfi±V^+W^ 

Subtracting  (1)  from  (4), 


2f=z  ^a^±Va*  +  ^b^; 

,                                     .   i/-  a"  ±  Va'  +  4^ 
whence,  y=:±y — -! . 

4.  Solve  the  equations 

2/2-^:2^16    .    .     .     (1), 

2y^  -  4a;y  +  3a;2  =  17    .     .     .     (2). 
Assume  y  =  vx;  then  (1)  and  (2)  become 
v2a>J  — a;2  =  16     .     .     .     (3), 


PABTICULAR    SYSTEMS.  263 

2v2a^  _  4vx^  _|.  sx^  =  17    .    .    .     (4). 
16 


From  (3),  x^  = 


1' 


and  from  (4),  ^2^__i|__; 


17  16 


(5). 


2v^-.^v-\-3      v^  —  1     ' 

Clearing  (5)  of  fractions,  transposing,  and  reducing, 

Uvu  —  642;  =  —  65    .    .    .    (6) ; 

6         13 
whence,  ^~q  ^^  ~^* 

5  16 

Substituting  -  for  v  in  the  two  equations  x^  =  -^ — -  and 

yz=vx,  we  have  x?  =  ^,  and  yz=-x', 

m 
x=  ±3    and    ^  =  ±  5. 

13 

Substituting  -^  for  v  in  the  same  equations,  we  find 

a^=±3    and    y=±Y' 

The  artifice  here  used  may  be  adopted  whenever  all  the  un- 
known terms  in  both  equations  are  of  the  second  degree  with  ref- 
erence to  the  unknown  quantities. 


5.  Solve  the  equations 

gfi  ^  xy  —  6y^  =  24:    .     .     . 

(1), 

x^ -{- 3xy  -  lOy^  =  32    .    . 

.    (2). 

Assuming  y  =  vx,   (1)  and  (2)  become 

x^  4-  vx^  —  6vW  =  24    .    . 

.    (3), 

ayi  ^3v(x^ --lOv^x^  =  32    .    . 

.     (4). 

264  SIMULTANEOUS    EQUATIONS. 

24 


From  (3),  x^  = 


1  4- V  — 6t^^ 


32 

and  from  (4),  ^,2,^  _____; 

24  32 


1  4-  v  —  6y2      1+  3i;  —  10t;2 

whence,  ^^  =  o    ^^   q  * 

2  3 


(5); 


1  24 

Substituting  ^  for  v  in  the  two  equations  x^  — ^ 

and  y  =  vx,  we  have  3?^  =  cx^,  and  y  =  -x; 
a;  =±00     and    y=:t^» 
Suostituting  ^  for  v  in  the  same  equations, 
^  a;  =  ±  6    and    ^  =  ±  2. 

6.  Solve  the  equations 

a^  +  2xy-\-y^-{^ax  +  ay  =  b    .    .    .     (I), 
xy  +  y^  =  c    .    .    .     (2). 

The  first  equation  may  be  written  thus : 

{x  +  yY-{-a{x  +  y)=zb; 


whence,        x  -\-  y  = .•    •    •    (3)- 

From  (2),  x-hy  =  -    .    .    •    (4). 

Combining  (3)  and  (4),  we  find 

a2^2b  —  2cTaV¥TU 


x  = 


—  a±  Va^+  46 
2c 


a  ±  ^Ai^  4-  42> 


PARTICULAR    SYSTEMS.  265 

7.  Solve  the  equations 

a?  —  2rr^  -\- y^  —  ax  ■]-  ay  =  b    .     .    .     (1), 
xy  —  y^  =  c    .    .    .     (2). 

Equation  (1)  may  be  written  thus: 

{x  —  yY  —  a(x  —  y)=h', 

whence,  x  —  y=z •    •    •    V^)* 

Prom  (3),  ''-y  =  l    •    •    •    W- 

Combining  (3)  and  (4),  we  find 


y 


_a^  +  2b-\-2c±a  Va^  +  ^b 
~~  a±  Va^  +  46 

2c 


a  ±  Va^  H-  45 
8.  Solve  the  equations 

xiy-\.xy^  =  30    .    .     .     (1), 

1+1=1  ■  •  •  (^)- 

Equation  (1)  may  be  reduced  to  the  form 

xy{x-\-y)=SO    ,     .    .     (3), 

and  (2)  may  be  reduced  to  the  form 

6(x  +  y)=6xy    .    .    .    (4). 

Dividing  (3)  by  (4), 

xy  _  30  . 
6  ~  6xy' 

whence,  6x^y^  =  180, 

or,  a;y  =  36; 

whence,  xy=  ±6    .    .    .     (5), 


266  SIMULTANEOUS    EQUATIOKSw 

Combining  (3)  and  (5), 

iP  +  y=±5    .    .    .     (6). 

Combining  (5)  and  (6),  we  find 

x  =  S,  2,      1,    or    —6, 
y  =  2,  3,  —6,     or         L 

9,  Solve  the  equations 

X  -hy  =a    .    .    .     (1), 

af^Jff  =  h^   .     .    .     (2). 
Dividing  (2)  by  (1), 

ic*  —  a^Sy  +  x^y^  —  xy^  -\- y^  = -', 

ct 

which  may  be  placed  under  the  form 

^^-y''-xy(xi^y^)-\-x^y'^  =  ^    .    .    .    (3). 

Squaring  (1)  and  transposing  the  term  2xy, 

a^-i-y^  =ai  —  2xy    .    .    .     (4). 
Squaring  (4), 

X^-\-2x^f-\-y*z=za*^^^y-\-4xiy^    .    .    .     (5); 
whence,      x*  -{-  y^=  a*  —  4:a^xy  +  2x^y^    .    .    .     (6). 

Equation  (3)  may  therefore  be  placed  under  the  form 
a^'^^aHy  ■\'2o?y'^  —  xy{a^  —2xy)  +  x^^-^ 

that  is,  hx^y'^-^aHy  —  ^'^a?'    ...     (7); 

Cb 


a^±y 


4:b^  +  a^ 


whence,  xy  = —^ —     .    .    .    (8). 

The  values  of  x  and  y  may  now  be  found  by  combining  (1) 
and  (8). 


PARTICULAK    SYSTEMS. 

Solve  the  following  pairs  of  equations  : 


267 


10.     i^  +  ^^  =  65) 


11. 


Ans.    )^=±^or  ±4, 
±  4  or  ±7. 


\y 


Ans.    (^=±3  or  qp  8, 


12. 


j  x^  +  Zxy=  54 ) 
\xy  +  4.y^  =115  4" 


x=  ±d  or  ±36, 

Ans.   \  23 

«/  =  ±  5  or  =F  y . 


13. 


(^2  +  ^^  =  15)  ^^^^ 


,  a;  =  ±  3  or  ±  -—, 
(  V2 


y=±2  or  ± 


V2 


I  3a^  4-  8?/2  =  14  ) 


a;=  ±2or    ±\/% 

0 


15.     i^+*y  =  12l.  j„,. 

(  a:?/  —  2?/*  =   1 ) 


(i/2_2a:?/  +  15=    0) 


(  a:^  +  2w2  =  3  i 


W5. 


-4W5. 


18. 


a;      y      5 
rrj^  OK  180 


a;  =  ±  3  or  ± 


^  =  ±  1  or  ± 


8 

1 


V6 

a:  =  ±  4  or  ±3^/3, 
2/  =  ±  5  or  ±     Vs. 

15 


x=± 


V21 
3 


or  i:  00, 


^       =^  V21 


Ans,    i^  =  30or6 
( 2/  =i  6    or  30. 


268 


SIMULTANEOUS    EQUATIOlfS. 


19. 


20. 


21. 


22. 


(  ic2  +  1/2  =  45 

i    ^  +  ^  =  ^[.  Ans, 

iSy  —  x  =  y^) 


.  Ans. 


]' 
U 


x=±6, 


±3  or  :F3. 


^  +  3^^=o^2' 


x  —  y 


xy 


x  =  0,  2,  or     ±  a/2, 
3^  =  0,2,  or  2T  A/^. 

.         ( a;  =  0,  4,  or  —  2, 
(  y  =  0,  2,  or  —  4. 


(3?^y^^x  +  y=\^) 
X  xy   ==    Q) 

A71S, 


(  a;  =  3,  2,  or  — 
(y  =  2,3,  or  - 


-3±V3, 
3:fV3. 


23.  \3^  +  y'-x-y  =  32) 
'  \x  -\-y  -\-xy       =29) 


^W5. 


rr  =  5,  4,  or  —  5  ±  V—  14, 
y  =  4:,  5,  or  —  5  =F  V—  14. 


24.  -1^+.^=  n 

(  a:3  +  y3  ,^3  65  f 


.         (  a;  =  4  or  1, 

Ans,    \         .        / 

\y  =il  or  4. 

( 2/  =  0  or  —  2. 


In  some  of  the  following  answers  the  roots  are  not  all  given 


26. 


{xy{x+y)=30\ 
{7?  + if       =  35  ) 


2:5  __^=:  3093 


28     i  ^-2^  =  3093) 
■ix  —y  —       3  ) 


Ans. 


\y  = 


x  =  3   or  2, 
2   or  3. 


27.     ■i!,  +  ^=i[-  ^^^. 

(a;4  +  «*  =  82i 


a:=3,  1,  or  2±5\/— 1, 
2/=l,  3,  or  2T5a/^1. 

^/i..    i^  =  5or  -2, 
(3/ =  2  or  —5. 


PAETICULAR    SYSTEMS. 


269 


29.  j^^-^y+2/'=in.     J, 

(  X  —  xy     -\-  y  ^=    4  ) 


30. 


32. 


33 


\x^  +  xY-\-y^  =  ^^^  ^' 

31       U2  +  2/2  +    :r^  :=  49  ) 
ix^  +  y^  +  xY=931  ) 

(a4_^^_y4_^2^84) 
(  2;2  +       a^i/2  +  «/2  =  49  f  • 

J  (2;2  —  y^){x  —y)  =  16a:y 
(  (V  _  ^)  (^2  _  y2)  =  UOx^y^ 

34    \(f<^  =  y{s-y)      I, 

I  y^+{s  —  yY  =  x^) 
Ans. 


Ans, 


Ans. 


Ans. 


2/=J(9TV73). 

{x=  ±3  or  ±2, 
U=±2  or  ±3. 


X=:  ±5 

or 

±3, 

y=±3 

or 

±5. 

x=  ±3 

or 

±3, 

y=±^ 

or 

±3. 

Ans,   \  ^=^  ^^  ^' 
( «/=3  or  9. 


y  = 


35. 


J  a;2^i  +  2/^^1  =  9  ) 


^^5. 


ic=:4  or  2, 
2/  =  2   or  4. 


PAIRS  OF  EQUATIONS  INVOLVING  RADICAL  QUANTITIES. 
393,  EXAMPLES. 

1.  Solve  the  equations 

V^  +  V^  =  5    .    .     .     (1), 
x  +  y  =  13    .     .    .     (2). 
Squaring  (1)  and  subtracting  (2)  from  the  result, 

2  Vxy  =  12 ; 
whence,  ^y  =  144    .    .    .    (3). 


270  SIMULTANEOUS    EQUATIONS. 

Squaring  (2)  and  subtracting  (3)  from  the  result, 
a^2-2a-^  + 2/2  =  25; 
whence,  a:  —  y  =  ±  5    .    ^    .     (4). 

ix  —  9  or  4 
y  =  4:  or  9! 

2.  Solve  the  equations 


Jx  —  y         20  .-V 

--y  +  V^|  =  ^T^  •  •  •   «> 

0^5  +  2/^  =  34    .     .     .     (2). 
From  (1),  by  transposition  and  reduction, 


^,y2^20^        /x-y    _         ,3x 

x^-y  ^  x  +  y 


Dividing  the  denominators  of  (3)  by  Vx-\-yy 


Vx-\-y 


=  —  Vx  —  y; 


whence,  a^  -  y"^ -20  =  -  Vx'-y^; 

or,  x^-y^-\-  Vx^-y^  =  20     .    .    .    (4). 


Assuming   \/Q?  —  y^  =  z,  (4)  becomes 

«a  +  ^  =  20    .    .    .    (5); 
whence,  2;  =  4  or   —5; 

that  is,  jr2-y2=i6  or  25    .    .    .     (6)/ 

fa;  =  ±  5  or    ± 
3 


5;9 

2' 


INVOLVING    RADICAL    QUANTITIES.  271 

3.  Solve  the  equations 

a;J  +  /=:35     .     .     .     (1), 

xi  +  y^=    5    .    .    .     (2). 

Assuming   a;*  =  v    and   y^  =  2,    (1)  and  (2)  become 
^3  +  ;23  =  35     .    .    .     (3), 

V  -{-  z  =6       ...     (4). 

These  equations  may  now  be  solved  as  Ex.  24,  Art  39^. 

4.  Solve  the  equations 


^i*^i=:k*' 

•    • 

.    (1), 

V^-'^  +  V^^  =  78    . 

.    • 

(2). 

Clearing  (1)  of  fractions,  it  becomes 

x  +  y  =  61  +  \/xy    . 

.    . 

(3). 

Equation  (2)  may  be  written 

V^(V^  +  V^)  =  78    . 

.    . 

(4). 

*  Assume                x  +  y  =  v    ,    .    . 

(5), 

and                               \/xy  =  z     .    ,    , 

(6). 

Multiplying  (6)  by  2  and  adding  the  result  to  (5), 

X  -\-2  Vxy  -\-  y  =  v 

+  2. 

> 

whence,         ^/x  -f  V^  =  ±  ^v  +  'Zz 

•  (7)- 

Hence  (3)  and  (4)  become 

t;=61  +  2j    .    .    . 

(8), 

^/z{±  Vv  +  22)  =  78 

•    (9). 

The  values  of  v  and  z  are  easily  found  from  (8)  and  (9),  and 
then  (5)  and  (6)  will  furnish  the  values  of  x  and  y. 


272  SIMULTANEOUS    EQUATIONS. 

Solve  the  following  pairs  of  equations : 


'-{'. 


-h  y  +  Vxy=    7 


=  21) 


+  r  +  xy 

+  ?/  —VI 

^^f^xy      =84 


^    U  +  ?/  —  Vxy  =    6 


.       (  a;  =  4  or  1, 
(  ^  =  1  or  4. 

\y  —  %   or  8. 


(a:+ Va^_3^2  =  8) 
^'    \x-y  =1) 


a;  =  13  or  6, 
12   or  4. 


a;  +  2^  =  10 


•    I         a;  4- 2/ =  20        i 


(y  =  2   or  8. 


-4/15. 


a;=10±4V6  or  10  T  ^Vl5, 
y=10:F4\/6  or  10  ±  Vl5. 


10. 


y  J 


.  Ans. 


a^c  —  2c±:ac  Va^  —  4 


y=± 


±2V^'^^ 
c 


11. 


a:y  —  (a;  +  2^)  =  54  ^ 


^TZS 


Vfl2-4 
(2/= 


6  or  —4 J, 
12  or  —9. 


12 


iC  =  ^2  ^  2 


a:  =  6   or  3, 
2   or  1. 


(2/  = 


MORE    THAif    TWO    Ui^KNOWJ^    QUANTITY. 


273 


13.  )^^^y^ 


14. 


j  ^  + y^ 


An8 


=  189 


{  X  -i-  y  +  ^x  +  ?/  =    12 


15.  :^^y  —  's/y  —  xz=^  Va  —  x 


16.   s^ 


Jx-\-y  6 

a:  +  w  —  i/  — i-^  = 

^    X  —  y      X  —  y 

iC2  4.  ^2  _  41 


-4?^s. 


J  a;  =  81    or  16, 
•  ( ?/  =    8   or  27. 

ic  =  5   or  4, 


^W5. 


or  5. 


Ans. 


4 


y  =  -^a. 


±5  or    ±3l/|, 


y  =  ±  4   or    ± 


^1 

2  ' 


GROUPS  WITH  MORE  THAN  TWO  UNKNOWN  QUANTITIES. 

394,  EXAMPLES. 

1  Solve  the  equations 

^  +  ^^  +  2/'-^  =  37  .  .  .  (1), 

x^-^xz  +  z^z^^^  .  .  .  (2), 

y^  +  yz  +  z^=:l^  .  .  .  (3). 

Subtracting  (2)  from  (1), 

{y^z)x  +  y^^z^  =  ^', 
whence  by  factoring, 

(y-z){x  +  y  +  z)z:^%    .     .     .     (4). 
Subtracting  (3)  from  (2)  and  factoring  the  result, 


{x-y){x-{-y  -^z)  =  ^ 


(5). 


18 


274  SIMULTANEOUS    EQUATIONS. 

Combining  (4)  and  (5), 

whence,  x  -{■  z  =  'Zy    .     .    .     (6). 

Combining  (5)  and  (6), 

(x-y)^y  =  ^', 

whence,  a;  =  -  +  ^    .    .    .    (7). 

if 

OomlDining  (1)  and  (7), 

(?  +  y)'+ 3 +  3^  +  ^2  =  37; 
whence,  ^  =  ±  3    or     ±  o  "V^- 

Substituting  these  values  in  (7), 

x=±i    or     ±yV3. 
Substituting  for  x  and  y  their  values,  we  find  from  (6) 
2  =  ±  2    or     =F  I V3. 

2.  Solve  the  equations 

x  +  y  +  z—a  .  .  .  (1), 

a?+y^+z^=l^  .  .  .  (2), 

xy=cz  .  .  .  (3). 

Transposing  z  in  (1), 

x-\-y  =  a  —  z; 

tyhence,     x^  +  2xy  +  y^  =  a^  ^  2az  +  2:2    .    .    .     (4^^ 

Transposing  z^  in  (2), 

a^^y2=:l2  —  zZ      .      .      .       (5). 

Subtracting  (5)  from  (4), 

2xy  =  a^  +  2z^^  2az  —  i^    .    .    .     (6). 


MORE    THAN    TWO    UNKNOWN    QUANTITIES. 


275 


Combining  (3)  and  (6), 

2cz  =  a2  ^  2z^  —  2az  —  h^ 


(7); 


whence, 


a  +  c  ±  V(a  4-  c)2  —  2  {a^  —  b^) 


2 


Substituting  these  values  for  2;  in  (1)  and  (3), 


a  +  c±V(a  +  c)^—2(a^—d^) 
^  +  y  + — —^^ ^ =  a 


(8), 


xy 


_ac-\-c^±cV(a  +  cy—2{a^—b^) 


(9). 


From  (8)  and  (9)  the  values  of  x  and  y  may  be  found. 
Solve  the  following  groups  of  equations : 

'  xy^s^  =  4725 

yz^ 45 

3.  \T  ~Y 
j___3 
x^y  ~  245 


X  -\-y  -^  z  =13 

X'i  -f-  ^2  ^  ;g2  _  61 

2yz  z=  X  {z -\- y) 


Ans. 


5. 


X 

a  + 

b" 

1 

X 

z 

-  + 

i 

a 

c 

y^ 

h 

c 

z 

. 

1     1  1    n 

-  H-  -  +  -  =  9 

X       y  z 

X      y 


(^  =  7, 

Ans.    Uj=:b, 

(z  =3, 

*  a;  =  9  or  4, 

•.    \  y  =  2±  V  — 14  or  3, 

\z=2'=f  V-14  or  6. 

ix=:0  or  2a, 

Ans,   \y  =  b  or  —b, 

z  •=.  c  or  —  c. 

r      1     6 

^=2^^2G' 

Ans,    - 

1        15 
y  =  3or^, 

1        15 
^  =  7  or  77. 
L          4        44 

y  -{•  z  = 
z  +  X 
X  -\-y  = 


SIMULTANEOUS    EQUATIONS. 
1 


X 

1 

y 
1 

z  J 


Arts,  x  =  y  z=zz=i  ±ro* 


x^  +xy  -{-  xz  =  27 

^y  +  y^  +yz  =  is 

xz  -\-  yz  -^  z^  =^  36 


U  =  ±4. 


10. 


y  = 

z    * 
"a; 

2;  = 

V 

v  = 

a 

'  X 

[bx= 

^vyJ 

^xyz  =  6 
xyv  =  8 
xzv  =  12 

L  yzv  =  24  J 


^W5. 


±  Va 


.^* 

y  =  V*, 

2;  =  ±  a/«, 

^v  =  ±  Va  Vb. 

r  x  =  i, 

Ans.    - 

y  =  3. 

0=3, 

w  =  4. 

395. 


PROBLEMS. 


1.  The  sum  of  two  numbers  is  equal  to  nine  times  their  differ- 
ence ;  and  if  the  greater  be  subtracted  from  their  product,  the  re- 
mainder will  be  equal  to  twelve  times  the  quotient  obtained  by 
dividing  the  greater  by  the  less:  find  the  numbers. 

Ans,  5  and  4. 

2.  The  sum  of  two  numbers  is  equal  to  a  times  their  differ- 
ence; and  if  the  greater  be  subtracted  from  their  product,  the  re- 
mainder will  be  equal  to  h  times  the  quotient  obtained  by  dividing 
the  greater  by  the  less:  find  the  numbers. 


-j^(i±^ 


and 


U/^M- 


PROBLEMS.  27? 

3.  Find  two  numbers  whose  difference  is  equal  to  two-ninths 
of  the  greater,  and  the  difference  of  whose  squares  is  128. 

Ans.  18  and  14. 

4.  Find  two  numbers  whose  difference  is  equal  to  -  of  the 
greater,  and  the  difference  of  whose  squares  is  a. 


5.  The  sum  of  two  numbers  is  16,  and  the  quotient  obtained 
by  dividing  the  greater  by  the  less  is  2J  times  the  quotient  ob- 
tained by  dividing  the  less  by  the  greater :  find  the  numbers. 

Ans,  10  and  6. 

6.  The  sum  of  two  numbers  is  «,  and  the  quotient  obtained  by 
dividing  the  greater  by  the  less  is  l  times  the  quotient  obtained 
by  dividing  the  less  by  the  greater :  find  the  numbers. 

0  —  1  0  —  \ 

7.  The  difference  of  two  numbers  is  15,  and  half  their  product 
is  equal  to  the  cube  of  the  smaller  number :  find  the  numbers. 

Ans.  18  and  3. 

8.  The  difference  of  two  numbers  is  d,  and  half  their  product 
is  equal  to  the  cube  of  the  smaller  number:  find  the  numbers. 

.     1  +  4^  ±  VsTTTi     .  1  ±  V8^  +  i 

Ans. ^ ■ —  and  —^ — ^ ■ — 

4  4 

9.  The  product  of  two  numbers  is  24,  and  the  product  of  their 
sum  and  their  difference  is  20 :  find  the  numbers. 

Ans.  6  and  4. 

10.  The  product  of  two  numbers  is  «,  and  the  product  of 
their  sum  and  their  difference  is  h'.  find  the  numbers. 


Ans,  ±  yb  ±  V^a^  +  ^  and  ±  ^  -h  ±  Vla^ -^l^^ 
2  2 


278  SIMULTANEOUS    EQUATIONS. 

11.  The  product  of  two  numbers  is  18  times  their  difference, 
and  the  sum  of  their  squares  is  117 :  find  the  numbers. 

Ans.  9  and  6. 
13.  The  product  of  two  numbers  is  ?;i  times  their  difference, 
and  the  sum  of  their  squares  is  a :  find  the  numbers. 

Ans,  c  ±  Vc  (2?n  +  c)    and    —  c  ±  Vc  (2//i  +  c), 

where  c  =  "  "^  ±  f  "+^. 

13.  Two  persons,  A  and  B,  bought  a  farm  containing  600  acres, 
for  which  they  paid  $600,  each  paying  1300.  A  paid  75  cents 
more  per  acre  than  B  in  order  to  be  permitted  to  take  his  share 
from  the  best  land.  How  many  acres  did  each  get,  and  at  what 
price  per  acre  ?    A7is.  A  200  acres  at  $1.50,  B  400  acres  at  $0.75. 

14.  A  certain  number  of  workmen  in  8  hours  carried  a  pile  of 
stones  from  one  place  to  another.  Had  there  been  8  more  work- 
men, and  had  each  cari'ied  each  time  5  pounds  less,  the  pile  would 
have  been  removed  in  7  hours ;  but  if  there  had  been  8  workmen 
less,  and  had  each  carried  each  time  11  pounds  more,  it  would 
have  required  9  hours  to  remove  the  pile.  How  many  workmen 
were  employed,  and  how  many  pounds  did  each  carry  at  a  time  ? 

Ans,  28  workmen,  and  each  carried  45. pounds; 
or  36  workmen,  and  each  carried  77  pounds. 

15.  The  fore- wheel  of  a  carriage  makes  6  revolutions  more  than 
the  hind-wheel  in  going  120  yards ;  but  if  the  circumference  of 
each  wheel  be  increased  one  yard,  the  fore-wheel  will  make  only  4 
revolutions  more  than  the  hind-wheel  in  going  the  same  distance. 
What  is  the  circumference  of  each  ? 

Ans,  Fore-wheel,  4  yds.;  hind-wheel,  5  yds. 

16.  What  number  is  that,  which  being  divided  by  the  product 
of  its  two  digits  gives  2  for  the  quotient,  and  if  27  be  added  to  it 
the  order  of  the  digits  will  be  inverted  ?  Ans.  36. 

17.  A  sets  off  from  London  to  York,  and  B  at  the  same  time 
from  York  to  London,  each  traveling  at  a  uniform  rate.  A  reaches 
York  16  hours,  and  B  reaches  London  36  hours  after  they  have 
met  on  the  road.  Find  in  what  time  each  has  performed  the 
journey.  A7is,  A  in  40  hours,   B  in  60  hours. 


SYNOPSIS    FOR    REVIEW.  279 

18.  A  man  had  $1300,  which  he  divided  into  two  parts,  and 
placed  at  interest  at  such  rates  that  the  incomes  from  them  were 
equal.  If  he  had  put  out  the  first  part  at  the  same  rate  as  the 
second,  he  would  have  drawn  for  this  part  136  interest;  and  if  he 
had  put  out  the  second  part  at  the  same  rate  as  the  first,  he  would 
have  drawn  for  it  $49  interest.     Find  the  rates  of  interest. 

Ans.  6  per  cent,  for  the  larger  part,  and 
7  per  cent,  for  the  smaller. 

19.  A  and  B  engage  to  reap  a  field  for  $24;  and  as  A  alone 
could  reap  it  in  9  days  they  promise  to  complete  it  in  5  days. 
They  found,  however,  that  they  were  obliged  to  call  in  0  to  assist 
them  2  days  in  order  to  complete  the  work  in  the  stipulated  time, 
in  consequence  of  which  B  received  $1  less  than  he  would  have 
done  if  he  and  A,  \nthout  the  assistance  of  C,  had  continued  until 
they  completed  the  work.  In  what  time  could  B  or  C  alone  reap 
the  field?  Ans.  B  in  15  days,  0  in  18  days. 

20.  A  number  consists  of  three  digits.  The  sum  of  the  squares 
of  the  digits  is  104;  the  square  of  the  middle  digit  exceeds  twice 
the  product  of  the  other  two  by  4 ;  and  if  594  be  subtracted  from 
the  number  the  order  of  the  digits  will  be  inverted:  find  the 
number.  Ans.  862. 

396.  SYNOPSIS   FOR   REVIEW. 


O 

t— I 

EH 

.< 
>^ 

^  H 

S5 


'  Groups  and  pairs. 

Degree  of  an  equation. 

General  form  of  an  equation  of  the  second  degree  in- 
volving two  unknown  quantities. 

Pairs  consisting  of   one   equation  of   the  first  degree 

AND    one    of    the    SECOND    DEGREE. 

Particular  systems. 

Pairs  of  equations  involving  radicals. 

Groups  involving  more  than  two  unknown  quantities 
and  one  or  more  equations  of  a  higher  degree  than 
the  first. 


OHAPTEE   XVL 
EATIO,  PEOPORTIOl^,  AJSTD  YAEIATION. 


RATIO. 

397.  Hie  Hatio  of  two  quantities  is  the  quotient  arising 
from  dividing  the  first  by  the  second.    Thus,  the  ratio  of  6  to  3 

is  2,  and  the  ratio  of  a  to  J  is  t.* 

398.  The  Sign  of  ratio  is  the  colon.  Thus,  a:b  ia 
read  the  ratio  of  a  to  b, 

399.  The  Terms  of  a  ratio  are  the  quantities  com- 
pared. Thus,  in  the  expression  a :  b,  the  terms  of  the  ratio  are 
a  and  b. 

400.  The  Antecedent  of  a  ratio  is  its  first  term,  and 
the  Consequent  of  a  ratio  is  its  second  term. 

The  antecedent  and  consequent  of  a  ratio  together  form  a 
Couplet, 

401.  A  Simple  Hatio  is  one  whose  terms  are  entire. 
Thus,  a:b  is  a  simple  ratio. 

*  Ratio  as  thus  defined  is  sometimes  called  geometrical  ratio  or  ratio  hy 
quotient,  to  distinguisli  it  from  arithmetical  ratio  or  ratio  hy  difference.  The 
arithmetical  ratio  of  a  to  &  is  a — &.  The  sign  of  arithmetical  ratio  is  •• . 
Thus,  a  --h  is  read  the  arithmetical  ratio  of  a  to  h. 

When  the  word  ratio  is  used  without  modification  it  is  understood  to 
mean  geometrical  rcuio. 


EATIO.  281 

403.  A  Complex  Matio  is  one  in  which  at  least  one  of 

the  terms  involves  a  fraction.     Thus,    -  :  b  and  -  :  -  are  complex 

c  c    a  ^ 

ratios. 

403.  A  CoTVipound  Hatio  is  the  ratio  of  the  products 
of  the  coiTesponding  terms  of  two  or  more  ratios.  Thus,  the  ratio 
compounded  of  a:b  and  c:d  is  ac:  bd. 

A  compound  ratio  does  not  differ  in  its  nature  from  any  other 
ratio.    The  term  is  used  to  denote  the  origin  of  the  ratio. 

404.  The  Duplicate  Matio  of  two  quantities  is  the 
ratio  of  their  squares.    Thus,  a^:l^  is  the  duplicate  ratio  of  a  to  b, 

405.  The  Triplicate  Matio  of  two  quantities  is  the 
ratio  of  their  cubes.    Thus,  a^ :  b^  is  the  triplicate  ratio  of  a  to  b. 

406.  Tlie  Subduplicate  Matio  of  two  quantities  is  the 
ratio  of  their  square  roots.  Thus,  ^/a  :  Vb  is  the  subduplicate 
ratio  of  a  to  b. 

407.  The  Subtriplicate  Matio  of  two  quantities  is 
the  ratio  of  their  cube  roots.  Thus,  ^s/a  :  Vb  is  the  subtriplicate 
ratio  of  a  to  b. 

408.  T7ie  Direct  Matio  of  two  quantities  is  the  quo- 
tient arising  from  dividing  the  antecedent  by  the  consequent. 

Thus,  the  direct  ratio  of  a  to  5  is  t- 

409.  Tlie   Inverse   or  Meciprocal   Matio   of   two 

quantities  is  the  direct  ratio  of  their  reciprocals,  or  the  quotient 
arising  from  dividing  the  consequent  by  the  antecedent.    Thus, 

the  inverse  ratio  of  a  to  5  is  -  :  ^  or  -. 

a  b        a 

410.  A  ratio  is  called  a  ratio  of  Greater  Inequality, 
of  TjCSH  Inequality 9  or  of  Equality,  according  as  the 
antecedent  is  greater  than,  less  than,  or  equal  to,  the  consequent. 


282 


PEOPOETIOK. 


411.  EXAMPLES, 

1.  Find  the  ratio  of  a^  —  h^  to  a  +  h. 

a  -\-  0 

2.  Find  the  inverse  ratio  of  a^  —  h^  to  a  ■\-  b.    Ans. t. 

a  —  h 

3.  Find  the  ratio  which  is  compounded  of  3  :  5  and  7  :  9. 


Ans, 


15 


4.  Find  the  subduplicate  ratio  of  100  to  144.  Ans.  -. 

6 

6.  Show  that  a ;  ^  is  the  duplicate  of  a  +  c  :  J  +  c  if  c^  =  a5. 


PROPORTION. 


413.  A  Proportion  is  an  equation  in  which  each  member 
is  a  ratio,  both  terms  of  which  are  expressed.* 

The  equality  of  the  two  ratios  may  be  indicated  either  by  the 
sign  =,  or  by  the  double  colon  : :.  Thus,  we  may  indicate  that 
the  ratio  of  8  to  4  is  equal  to  that  of  6  to  3  in  any  of  the  following 
ways: 

r8:4  =  6:3 
8:4::6:3 
.   8_6 
4~"3 
8-T-4  =  6-^3^ 

This  proportion,  in  any  of  its  forms,  is  read  tlie  ratio  of  8  to  U 
is  equal  to  the  ratio  of  6  to  3,  or,  8  is  to  Jf  as  6  is  to  3. 

*  A  geometrical  proportion  is  one  in  which  the  ratios  are  geometrical. 

An  arithmetical  proportion  is  one  in  which  the  ratios  are  arithmetical. 
Thus,  6  ••  5  —  9  ••  8  is  an  arithmetical  proportion. 

When  the  word  proportion  is  used  without  modification,  it  is  understood 
to  mean  geometrical  proportion. 


PEOPORTIOK.  283 

413.  Tlie  Terms  of  a  proportion  are  the  four  quantities 
which  are  compared.  The  first  and  second  terms  form  the  first 
couplet ;  the  third  and  fourth,  the  second  couplet. 

414.  The  Antecedents  in  a  proportion  are  the  first  and 
third  terms. 

415.  Tlie  Consequents  in  a  proportion  are  the  second 
SLud  fourth  terms. 

416.  The  Extremes  in  a  proportion  are  the  first  and 
fourth  terms. 

417.  The  Means  in  a  proportion  are  the  second  and  third 
terms. 

■  418.  If  four  quantities  a,  h,  c,  and  d  are  so  related  that 
a'.h  ^=c'.d, 
d  is  said  to  be  a  Fourth  Proportional  to  a,  h,  and  c, 

419.  If  three  quantities  «,  i,  and  c  are  so  related  that 

a-.h  =  h:Cf 
c  is  said  to  be  a  Third  JProjwrtional  to  a  and  b. 

420.  K  three  quantities  «,  J,  and  c  are  so  related  that 

a:b  ^b:c, 
b  is  said  to  be  a  Mean  Proportional  between  a  and  c, 

431.  A  Continued  Proportion  is  a  continued  equa- 
tion in  which  each  member  is  a  ratio,  both  terms  of  which  are 
expressed.  Thus,  a :  b  =  c :  d  =  e  :f  :=  g  :  h  is  a  continued  pro- 
portion. 

422.  If  four  quantities  a,  b,  c,  and  d  are  so  related  that 


a\b  —  ~.-y 


1.1 

c'd' 


they  are  said  to  be  Inversely  or  Iteciprocally  Propor- 
tional, 


284  PEOPOKTION. 

433.  JEquitnultiples  of  two  or  more  quantities  are  the 
products  obtained  by  multiplying  each  of  them  by  the  same  quan- 
tity.   Thus,  ma  and  mb  are  equimultiples  of  a  and  i. 

424.  A  proportion  is  taken  by  Alternation  when  the 
means  or  the  extremes  are  made  to  exchange  places.  Thus,  if 
a  :  b=c :  d,  we  have  by  alternation,  either  a  :  c^=b  :d,or  d'.  bz^c-.a. 

425.  A  proportion  is  taken  by  Inversion  when  the  terms 
of  each  couplet  are  made  to  exchange  places.  Thus,  if  a :  5  = 
c : df  we  have  by  inversion,  b:a=.  d'.c. 

426.  A  proportion  is  taken  by  Composition  when  the 
sum  of  the  terms  of  each  couplet  is  compared  with  either  term  of 
that  couplet,  the  same  order  being  observed  in  the  two  couplets. 
Thus,  if  « :  5  =  c :  i/,  we  have  by  composition,  either  a  -{-  b'.a  =z 
c  +  d:c,  OT  a  -\-  b:b  =  c  -\-  did. 

427.  A  proportion  is  taken  by  Division  when  the  differ- 
ence of  the  terms  of  each  couplet  is  compared  with  either  term  of 
that  couplet,  the  same  order  being  observed  in  the  two  couplets. 
Thus,  if  a:b  =  c:df  we  have  by  division,  either  a—b:a=c—d:Cj 
or  a  —  b'.b  ^  c  —  d:d. 

428.  In  every  proportion  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means. 

Let  a:b  =  c:d;  . 

then  |:=.|(412); 

whence,  ad  =  be. 

Cor. — If  the  means  are  equal,  as  in  the  proportion  a:b  =  b:c, 
we  have  b"^  =  ac,  whence  b  =  Vnc;  that  is,  a  mean  propor- 
tional between  two  quantities  is  equal  to  the  square  root  of  their 
product. 

429.  If  the  product  oftivo  quantities  is  equal  to  the  product 
of  two  others,  either  two  may  be  made  the  extremes,  and  the  other 
two  the  means,  of  a  proportion. 


Let 


PROPOBTION. 

ad=zlc    .    .    .    (1) ; 


285 


then,  dividing  both  members  by  cd, 

-=— ,    or    a\c=z  o:d    .    .    . 
c       d 

In  like  manner  it  may  be  shown  that 

a\h  =^  c  \d    .    .    .     (3), 


(2). 


c  \a-=.  d'.h 
c  \d=.  a'.h 
d'.h  z=  c\a 


(4), 
(5), 
(6), 


and  so  on. 

CoE.  1. — Any  one  of  these  proportions  may  be  inferred  from 
any  other.     Thus,  from  (2), 

ad  =  be, 

from  which  any  one  of  the  proportions  may  be  derived. 

Cor.  2. — Since  (3)  may  be  derived  from  (2),  it  follows  that 

If  four  quantities  are  in  proportion,  they  will  he  in  proportion 
hy  alternation. 

Cor.  3. — Since  (4)  may  be  derived  from  (2),  it  follows  that 

If  four  quantities  are  in  proportion,  they  will  he  in  proportion 
hy  inversion. 

430.  Equimultiples  of  two  quantities  are  in  the  same  ratio 
as  the  quantities  themselves. 


ma 
mh 


a 


ma:mh  =  a:  h. 


Cor.— K 


m  =  l± 


P 


we  have 
that  is, 


a  +  -a'.h  ±-b  =  a:h; 


286  PROPORTION. 

If  tico  quatitities  he  increased  or  diminislied  hy  like  parts 
of  each,  the  results  will  he  in  the  same  ratio  as  the  quatitities 

themselves. 

« 

431.  Any  equimultiples  of  one  couplet  of  a  proportion  are  in 
the  same  ratio  as  any  equimultiples  of  the  other  couplet. 

Let  a\h  =  c'.d\ 

then 

whence, 


a 

V 

c 

-d' 

ma 
mb 

nc 

ma :  mh 

=  nc :  nd. 

I   and 

n  =  l±K 

CoE. — ^If  m  =  l  ±-   and   w  =  1  ±  =^,  we  have 
^  <1 

a±^a:b±^b=ic±^c:d±^d; 
,      .  9  9  9  9 

that  IS, 

If  tJie  terms  of  the  first  couplet  of  a  proportion  he  increased  or 
diminished  hy  like  parts  of  each,  and  if  the  terms  of  the  second 
couplet  he  increased  or  diminished  hy  any  other  like  parts  of  each, 
the  results  will  be  in  proportion. 

433,  Any  equimultiples  of  the  antecedents  of  a  proportion 
are  in  the  same  ratio  as  any  equimultiples  of  the  consequents. 


Let                      a'.h=zc:d    .    .    . 

(1); 

then                                %^%    .    .    . 
b      a 

(2). 

Multiplying  (2)  by  ^, 

ma      mc 
nh  ""  nd' 

whence,  ma :  nh  =  mc :  7id, 

which  by  alternation  becomes 

ma:mG=:^nb:  nd. 


PROPOKTioi^r.  287 

433.  Axiom. — Ratios  that  are  equal  to  the  same  or  equal 
ratios  are  equal  to  each  other. 

Thus,  if  Q,'*l)  ^=  c:  d, 

e\f—g\h, 

and  a\b  =.e\f', 

then  c  I  d  =g\h. 

434.  If  the  ratio  of  the  antecedents  of  one  proportion  is  equal 
to  the  ratio  of  the  antecedents  of  another  proportion,  the  ratio  of 
the  consequents  of  the  one  will  he  equal  to  the  ratio  of  the  conse- 
quents of  the  other,  and  conversely. 

Let  a\l  =^  c  d    .    .    .    (1), 

e'.f=g:h    .    .    .    (2), 

and  a'.c=e'.g    .    .    .    (3) ; 

then  will  h  :  d  =/ :  h. 

For,  from  (1),  a\c  —  h\d  (429,  Cor.  2), 

and  from  (2),  e :  g  =/:  h ; 

b:d=f:h  {'^SS). 
In  hke  manner  it  may  be  shown  that,  if 

a'.h=:c\d, 

e\f  —  g'.h, 
and  'b\d^=.f'.h\ 

then  will  a'.c  =  e:g. 

435.  If  four  quantities  are  in  proportion,  they  will  he  in 
proportion  hy  composition  or  division. 

Let  a'.h  =  c'.d    .    .    .     (1), 

then  1  =  1    ••    •    (2)5. 

whence,  |  ±  1  =  |  ±  1, 


288                                                 PROPORTION". 

a±h      c±d 
b     =     d-    '    ' 

.  0^); 

whence,             a  ^^h  '.h  =  c  ±d:  d    . 

.    .     (4), 

Dividing  (3)  by  (2), 

a±l}      c±d 
a              c 

.   (5); 

whence,             a±d:a  =  c±6?:o    . 

.    .    (6). 

Cob.— Separating  (6), 

a  -^  b  '.  a=^c  -{-  d:  Cj 

and  a  —  b'.a=:ic  —  d'.c; 

a-\-b',a  —  b=iC  +  c? :  c  —  a?  (434); 

that  is, 

If  four  quantifies  are  in  proportion,  the  sum  of  the  first  and 
second  is  to  their  difference  as  the  sum  of  the  third  and  fourth  is 
to  their  difference. 

436.  Tlie  sum  of  any  number  of  the  an  tecedents  of  a  continued 
proportion  is  to  the  sum  of  the  corresponding  consequents  as  any 
antecedent  is  to  its  consequent. 

Let  .  a\b  =:  c \ d  =.  e \  f  z:^ g \h ^  &c., 

and  let  r  denote  the  ratio ;  then 

a      c       e      a      o^ 

whence,        a  =  br,    c  =  dr,    e  =  fr,    g  =  hr,    &c. ; 
whence,  by  addition, 

a  +  c  +  e+^  +  &c.  =  (J+  d  ■\-f+h-\-  &c.)r; 
whence       a±cj±_e_^r_g_±&^  _  ^ _ ^ _  £  _  &_ 

.-.    a  +  c  +  e+^  +  &c.  iS  +  c^+Z  +  A  +  Ac.  =za'.b  =c:d  =  &Q. 


PKOPORTION.  289 

437.  The  products  of  the  corresponding  terms  of  two  or  more 
proportions  are  in  proportion. 


Let 

a\h  =^c'.d, 
e\f  =g\hy 

and 

m\n  =p:q; 

then 

a      c 
b-d' 

«  _^ 
f-h' 

and 

m      p 
n-q' 

whence, 

by  multiplication. 

aem       cgp  ^ 
bfn  ~  dhq' 

aem  :  bfn  =  cgp :  dhq. 

438.  Tlie  quotients  of  the  corresponding  terms  of  two  propor- 
tions are  in  proportion. 


Let 

a:b  =ic:d. 

and 

e'.f  =  g\h\ 

then                   ' 

a      c 
l~d' 

and 

-f-v 

whence, 

af      ch 
te~dg' 

or, 

a      d  _c      b  ^ 

e      h      g     f 

« 

a    b       c    d 

19 


290  proportio:n^. 

439.  Like  powers  or  like  roots  of  the  terms  of  a  proportion 
are  in  proportion. 

Let  a:i  =  c:d    .    .    .     (1), 

then  l='a    ■    ■    ■    (')■ 

Kaising  (2)  to  the  n^  power, 


a" :  i"  =  c" :  <?». 
Extracting  the  w**  root  of  (2), 

^/a  _  Vc  ^ 
Vl'^Vd 

n  /—     n  /T-         n  /-     n 


Va  :  V^  =  Vc  :  VS. 


440.  TJtOBZEMS. 

1.  Find  two  nnmhers,  the  greater  of  which  shall  he  to  the  less 
as  their  sum  is  to  21,  and  as  their  difference  is  to  3. 

Let  X  =  the  greater  number,  and  y  =  the  less;  then  by  the 
conditions  of  the  problem. 


{x'.y  =  x-\-y:21    . 
\x:y  =  x  —  y  :    3     . 

.    .    (1)  L 
.     .     (2)  P 

\%lx  =  xy  +  y'^    .    . 
whence,              i    ^                 \ 

'                {    Zxz=xy  —  y^    ,    , 

.    (3)  L 

.  (4)  r 

Adding  (3)  and  (4), 

24:xz=2xy    .    . 

.   (5); 

whence,                                 y  =  12. 

Substituting  12  for  ?/  in  (4-)  we  find 

a:  =  16. 

PROPORTION.  291 

2.  Divide  the  number  14  into  two  such  parts  that  the  quotient 
arising  from  dividing  the  greater  by  the  less  shall  be  to  the  quo- 
tient arising  from  dividing  the  less  by  the  greater  as  16  is  to  9. 

Let  X  =  the  greater  number,  then  will  14  —  a?  =  the  less. 

By  the  conditions  of  the  problem, 

jj^:M^  =  16:9..    .    .    (1). 

Multiplying  the  terms  of  the  first  couplet  by   (14  —  x)x^ 
a:2:(14~a:)2  =  16:9     ...     (2)     (430); 
•whence,  rr:  14  — a:  =  4:3    .    .    .     (3)     (439). 

From  this  proportion  we  find 

a;  =  8    and    14  —  a:  =  6. 

3.  The  product  of  two  numbers  is  112,  and  the  difference  of 
their  cubes  is  to  the  cube  of  their  difference  as  31  is  to  3.  What 
are  the  numbers  ? 

Let  a;  =  the  greater  number,  and  y  =  the  less ;  then  by  the 
conditions  of  the  problem, 

\  xyr^lVl    .    .    .     (1)) 

(a^-;y3:(a;-j^)3  =  31:3    .    .    .     (2)) 

From  (2), 

a-2  +  a:?/  +  2/^:a:2_2a;2/  +  2/2  =  31:3     .     .     .     (3)  (430) ; 
whence,  ^xy\i^x  —  y)^  —  %%\Z    .    .    .     (4). 

Combining  (1)  and  (4), 

336:  (a: -2/)^  =  28: 3    .    .    .    (5); 
whence,  a:—?/ =6    .    .    .    (6). 

Combining  (1)  and  (6),  we  find 

xz=U,    2/  =  8. 


292  PEOPOKTIOIT. 

4.  What  two  numbers  are  tliosc  whose  difference  is  to  their 
sum  as  2  is  to  9,  and  whose  sum  is  to  their  product  as  18  is  to  77  ? 

Let  ic  =  the  greater  number,  and  y  =  the  less ;  then  by  the 
conditions  of  the  problem, 

Sx-y:x-^y=    2:    9     .    .    .     (1)) 
\x  +  y:xy       =18:77    .    .    .     (2)  f 

From  (1), 

2a::  2^  =  11:  7    .    .    .    (3)  (435,  CoE.); 

7ic 
whence,  y  =  —    .    .    .     (4). 

Combining  (2)  and  (4), 

J.  TjX  iX  O/^HJIW  /  ^\ 

_:-  =  18:77    .    .    .    (o); 

whence,  a:  =  11. 

Substituting  in  (4),  we  find 

5.  Two  numbers  have  such  a  relation  that  if  4  be  added  to 
each,  the  results  will  be  in  the  ratio  of  3  to  4 ;  and  if  4  be  sub- 
tracted from  each,  the  results  will  be  in  the  ratio  of  1  to  4.  What 
are  the  numbers?  A7is,  5  and  8. 

6.  Divide  the  number  27  into  two  such  parts  that  their  pro- 
duct shall  be  to  the  sum  of  their  squares  as  20  is  to  41. 

Ans.  12  and  15. 

7.  Two  numbers  are  in  the  ratio  of  3  to  2.  If  6  be  added  to 
the  greater  and  subtracted  from  the  less,  the  results  will  be  in  the 
ratio  of  3  to  1.     What  are  the  numbers?  A7is.  24  and  IG. 

8.  The  number  20  is  divided  into  two  parts  which  are  to  each 
other  in  the  duplicate  ratio  of  3  to  1.  Find  the  mean  propor- 
tional between  these  parts.  A7is.  G. 

9.  The  sum  of  the  cubes  of  two  numbers  is  to  the  difference 
of  their  cubes  as  559  is  to  127,  and  the  product  obtained  by  mul- 
tiplying the  less  by  the  square  of  the  greater  is  equal  to  294. 
What  are  the  numbers  ?  Ans,  7  and  G. 


VARIATION.  293 

10.  The  cube  of  the  first  of  two  numbers  is  to  the  square  of  the 
second  as  3  is  to  1,  and  the  cube  of  the  second  is  to  the  square  of 
the  first  as  96  is  to  1.     What  are  the  numbers  ? 

Ans.  12  and  24. 

11.  Given  m  -^  x:n-\-  x  =p  ^  x'.q  -\-  x  to  find  x. 

Ans.        ""P  -  '"^ 


m  -\-  q  —  n  —  p 
VARIATION. 

441,  One  quantity  varies  directly  as  another  when  the  two 
quantities  have  such  a  relation  that  one  increases  or  decreases  in 
the  same  ratio  as  the  other.  Thus,  in  uniform  motion,  the  space 
described  varies  directly  as  the  time. 

Sometimes,  for  brevity,  we  omit  the  word  directly,  and  say- 
simply  that  one  quantity  varies  as  another. 

442,  The  Sign  of  variation  is  the  symbol  oc .  Thus,  the 
expression  5  x  ^  is  read  s  varies  as  t. 

443,  A  Variation  consists  of  two  expressions  connected 
by  the  sign  oc . 

When  a  variation  immediately  follows  the  word  let,  the  sign  a  is  equiv- 
alent to  the  word  vari/. 

444,  One  quantity  varies  inversely  as  another  when  the  first 
varies  as  the  reciprocal  of  the  second.  Thus,  in  uniform  motion, 
if  the  space  (s)  is  constant,  the  time  (t)  varies  inversely  as  the  ve- 
locity (v) ;  that  is,   tec-. 

445,  One  quantity  varies  as  two  others yozw%  when  the  first 
varies  as  the  product  of  the  others.  Thus,  in  uniform  motion, 
the  space  (s)  varies  as  the  time  (t)  and  velocity  {v)  jointly ;  that 
is,  s  oc  vt. 

446,  One  quantity  varies  directly  as  a  second  quantity  and 
inversely  as  a  third,  when  the  first  varies  as  the  quotient  obtained 
by  dividing  the  second  by  the  third.     Thus,  in  uniform  motion. 


294  VARIATION. 

the  time  (/)  varies  directly  as  the  space  (s)  and  inversely  as  the 
velocity  (v) ;  that  is,  ^  oc  -. 

447.  If  one  quantify/  varies  as  anofJier,  either  of  them  is  equal 
to  the  product  obtained  by  multiplying  the  other  by  some  constant 
quantity. 

Let  ^  ^  yt 

and  suppose  y  =  J  when  a;  =  a ;  then 

x:a=y:b  (441) ; 

whence,  x  =  -^y  =  my, 

where  m  is  equal  to  the  constant  t- 

448.  If  one  variable  quantity  is  equal  to  the  product  obtained 
by  multiplying  another  by  a  constant,  the  first  varies  as  the  second. 

Let  X  =  my, 

and  suppose  y^b  when  x=z  a;  then 

a=zmb; 

x:a^=y:b; 

that  is,  X  cc  y. 

449.  If  one  quantity  varies  as  a  second,  and  the  second  as  a 
third,  the  first  varies  as  the  third. 

Let  i=^«n 

whence,  x  =  mnz ; 

xa:z     (448). 

450.  Jf  each  of  two  quantities  varies  as  a  third,  their  sum, 
their  difference,  or  their  mean  proportional  varies  as  the  third. 


VARIATIOK.  295 

Let  fn; 


then  ix  =  mz, 

\tj  =  nz  j 
whence, 

X  +  y  z:^  {m  -\-  n)z,    x  —  y  =  (m  —  n)z,    and     ^/xy  =  z  Vmn ; 
X  -^  y  a:  Zf        x  —  y  cc  z,        and         Vxy  oc  z. 

451.  i/*  owe  quantity  varies  as  tioo  others  jointly ,  either  of  the 
latter  varies  directly  as  the  first  and  inversely  as  the  other. 

Let  X  ccyz; 

then  X  =  myz ; 

\      X  1      i?" 

whence,  y  z=—.-,  and  z  =  —.-i 

^       m  z  my 

X  X 

y  a  -,  and  2;  oc  -. 

^      z'  y 

452.  If  one  quantity  varies  as  another,  either  varies  as  any 
multiple  of  the  other. 

Let  X  ^y'y 

then  x  =  my  =  —.ny; 

xccny. 
Again,  since  x  =  my, 

1  1 

^      m  mn 

yccnx. 

453.  If  both  members  of  a  variation  be  multiplied  or  divided 
by  the  same  quantity,  the  results  loill  vary  as  each  other. 

Let  X  cay        .     .     .     (1) ; 

tlien  x=zmy    .    .     .     (2). 


296 
Multif 

)lying  (2)  by  z. 

VAEIATIOISr. 

XZ  =  nizy\ 

Dividi 

ng  (2)  by  z. 

xz  oc  zy, 

X          y 

z          z' 

•*• 

X     y 

z      z 

Cor. — Hence  a  variation  may  be  cleared  of  fractions  in  the 
same  way  as  an  equation. 

454.  The  product  of  the  first  members  of  two  or  more  varia- 
tions varies  as  the  product  of  their  second  members. 

(xccy) 


Let 

^cc.     ; 

{  tuccv  ) 

then 

(x=my) 
]  t=nz   [; 

{iv  =  av  ) 

whence, 

xt2v  =  amnvyz'y 

.*. 

xttv  oc  vyz. 

455*  The  quotient  of  the  first  members  of  two  variations  varies 
as  the  quotient  of  their  second  members. 


Let 


{:=:/h 


X      m   y 
whence,  -  =  —  .  f  5 

'  z       n    t 


x      y 

Z       t 


VARIATION. 


297 


4:56,  LilcG  j)owers  or  like  roots  of  the  members  of  a  variation 
vary  as  each  other. 


Let 
then 
whence, 


xccy, 

X  =  my ; 

af»  =  m^y%  and   V^  =  V^  Vy ; 
ic»  a  y%        and   ^/x  oc  V^. 


457. 


PROBLEMS, 


1.  If  y  a:x  and  y  =  3   when  x  =  1,  what  is  the  value  of 
y  when   cc  =  3  ?  ^?2S.  9. 

2.  If  a;  oc  ?/  and  a;  =  15  when  ?/  =  3,  what  is  the  value  of 

y  in  terms  of  a;  ?  .  x 

^  A  ns.  y  =^  T- 

3.  If  zee  xy  and  z  =  l   when  a;  =  ?/  =  1>  """l^^^  is  the  value 
of  2J  when   .t  =  ?/  =2?  ^?Z5.  4. 

4.  If    zee px  +  ?/,    and  if    ;$;  =  3    when    a;  =  1    and   y  =  2, 
and  2=5   when  a;  =  2  and  y  =  3,  what  is  the  value  of  p  ? 

^ws.  1. 

5.  If  x^ccy^  and  a;  =  2  when  y  =  3,  what  is  the  value  of 

y  in  terms  of  a;  ?  3   -^ 

^W5.  2/  =  3  |/  — • 


G.  If 


^  oca; 
1 

X 


,  and,  if  1/  =  4  when  a:  =  1,  and  y=5 


when  a;  =  2,  what  is  the  value  of  y  in  terms  of  a;  ? 

2 
A  ns.  y  =  2x  +  -. 

X 

7.  If  xccy  when  2;  is  constant,  and  if  xazz  when  y  is  con- 
stant, how  does  x  vary  when  both  y  and  ;?  are  variable  ? 

^W5.  a:  X  ^jj. 


i>SS 


VARIATIOIT. 


458. 


SYNOPSIS    FOR    REVIEW. 


'  EATIO 


PEOPORTION. 


L  VAEIATION. 


I 


Terms. — Antecedent. — Consequent.— Couplet. 

Simple.  — Complex.  — Compound. 

Duplicate. — Triplicate. — Sub-dup. — Sub-trip. 

Direct. — Inverse  or  Reciprocal. — Of  greater 
IN  EQUAL. — Of  less  inequal. — Of  equality. 

Geometrical.  —  Arithmetical.  —  How  indi- 
cated?    How  READ? 

Terms  —  Antecedent.  —  Consequent.  —  Ex- 
tremes.— Means. 

Fourth  proportional. — Third  proportional. 
— Mean  proportional. 

Continued  proportion. 

Inverse  or  reciprocal  proportion. 

Equimultiples.  —  Alternation. —  Inversion. — 
Composition. — Division. 


Cor. 
Cor.  1, 
Cor. 
Cor. 


Cob. 


42§. 

429.    Cor.  1,  2,  3. 
4:iO, 
431. 
4:{2, 
Propositions.  ^  433, 

434. 

4:{5. 
436. 
437. 
438. 

439. 

'  Direct  variation. 
Sign  of  variation. 
A  variation. 
Inverse  variation. 
Joint  variation. 
Direct  and  inverse  variation  combined. 

447. 

44§. 

449. 

450. 

451. 

452. 

453.    Cob. 

454. 

455. 

450. 


Propositions. 


CHAPTER   XYII. 
MATHEMATICAL    Ilv^DUCTIOK 


459.  Mathematical  Induction  or  Demonstra^ 
tive  Induction  may  be  thus  described :  We  prove  that  if  a 
theorem  is  true  in  one  case,  whatever  that  case  may  be,  it  is  true 
in  another  case  which  we  may  call  the  7iext  case ;  we  prove  by 
trial  that  the  theorem  is  true  in  a  certain  case ;  hence  it  is  true 
in  the  next  case,  and  hence  in  the  next  to  that,  and  so  on ;  hence 
it  must  be  true  in  every  case  after  that  with  which  we  began. 
This  method  of  reasoning  is  exemphfied  in  the  demonstration  of 
the  following  theorems : 

460.  The  sum  of  n  consecutive  integers  hegitining  with  1  is 
n  {n  4-  1) 

2 
We  see  that  this  theorem  is  true  in  some  cases ;  for  example, 

1  +  2  =  ^ti),  1  +  3  +  3  =  iii+il ;   we  wish,  however, 

to  show  that  the  theorem  is  true  universally. 

Suppose  the  theorem  were  known  to  be  true  for  a  certain 
value  of  n ;  that  is,  suppose  for  this  value  of  n  that 

1  +  2  +  3  +  4+..  ..+«  =  ^"^^    .    .    .     (1). 

Adding  n  -\-\   to  both  members  of  (1), 

fi  (n  4-  1  ^ 

1  +  2  +  3  +  4+..  .+7j  +  7i-fl  =  -^^-^  +  n  H-  1  = 


(.  +  i)[l!L±|I±i]   .    .    . 


(3). 


300  MATHEMATICAL    INDUCTION. 

Therefore,  if  the  sum  of  n  consecutive  integers  beghming  with 

^i  (fi  _i_  1) 
1  is       ^       — -,    the  sum  of    w  +  1     such    numbers  will    bo 
Z 

(/I  _[- 1)    ^^       .J  •    Ii^  other  words,  if  the  theorem  is  true 

when  71  is  a  certain  number,  whatever  that  number  may  be,  it  is 
true  when  we  increase  that  number  by  1.  But  we  have  seen  by 
trial  that  the  theorem  is  true  when  7i  =  3 ;  it  is  therefore  true 
when  «  =  4 ;  it  is  therefore  true  when  ji  =  5;  and  so  on.  Hence 
the  theorem  must  be  universally  true. 

461.  Tlie  difference  between  the  like  poiuers  of  any  tivo  quan- 
tities is  divisible  by  the  difference  between  the  quantities. 

Let  a  and  b  denote  any  two  quantities,  and  let  n  be  any  posi- 
tive integer ;  then  will  «"  —  b^  be  divisible  by  a  —  b. 

«  ^  an-1  ^  b^±—-J—l      .      .      .       (1); 


a  —  d  a  —  b 

hence  a"  —  b^  is  divisible  by  a  —  b,  if  a"~^  —  Z*""^  is  divisible 
by  it.  Now  a  —  b\s  divisible  by  a  —  b\  therefore  a^  —  1?  is 
divisible  by  a  —  b\  therefore,  again,  a^  — -  b^  is  divisible  by  a—b, 
and  so  on ;  hence  a"  —  i"  is  divisible  by  a  —  b,  if  n  is  a  posi- 
tive integer. 

Performing  the  division  indicated  in  (1),  we  obtain 

Cor.  1. — The  number  of  terms  in  the  quotient  is  n. 

Cor.  2. — If  b  =  a,  each  term  of  the  quotient  becomes  equal 
to   a**"!;  hence, 

(?^l.=--  •  •  •  (^)- 

Cob.  3. — Substituting  (^  for  a  and  c^  for  b  in  (2),  we  have 

C2» ^n 

^    _^3    =  c2«-24-c2/i-4^_j.  ....  4.  c2^»-4^^«-2  .       .      .    (4). 


SYi?^OPSIS    FOR    EEVIEW.  301 

hence, 

Tlie  difference  hetween  the  like  evenpoivers  of  any  tivo  quan- 
tities is  divisible  by  the  difference  between  the  squares  of  the 
quantities, 

OoB.  4. — Multiplying  both  members  of  (4)  by  c  —  dy  we 
obtain 

^~f^  ={c  —  d)  (c2^-2  +  c^-^d^  4-  .  .  .  .  _j_  c'dJ^-i  -I-  d^-^) ; 
c  ~\~  a 

hence, 

TJie  difference  hetween  the  like  even  powers  of  any  two  quan- 
tities is  divisible  by  the  sum  of  the  quantities. 

Cor.  5. — Substituting  c^  for  a  and  d"^  for  b  in  (2),  we  have 

pinn flrnn 

Qin    _  dm 

hence, 

The  difference  between  the  like  powers  of  any  two  quantities  is 
divisible  by  the  difference  betiveen  any  other  like  powers  of  the  two 
quantities,  if  the  exponent  in  the  first  set  of  powers  is  divisible  by 
that  in  the  second  set. 

CoE.  6. — When  7i  is  odd,  we  have 

a  —  (—b)   ~   a  +  b' 
Now  by  the  theorem,  a"  —  (—  i)"    is  divisible  by    a  —  {—b); 
hence  a"  -|-  5"  is  divisible  hy  a  -{-  b  when  n  is  odd ;  that  is, 

T/ie  sum  of  the  like  odd  powers  of  any  two  quantities  is  divisible 
by  the  sum  of  the  quantities. 

Performing  the  division  indicated,  we  obtain 

a  ■\-  0 

462.  SYNOPSIS    FOR    REVIEW. 

CHAPTER    XVII.  (  (§60. 

MATHEMATICAL   INDUOTION.  (  ^^^^^^^^-  \  461.  Cor.  1,  2,  3, 4, 5, 6 


CHAPTEE   XVIII. 

PERMDTATmS-COMBIXATIOXS-BINOMIALFOMUU-EXTRACTIO.N' 
OF  HIGHER  ROOTS. 


PERMUTATIONS. 


463.  Tlie  Per^nutations  of  n  things,  taken  r  at  a  time, 
ai-e  the  results  obtained  by  arranging  the  things  in  every  possible 
order  in  groups  of  r  each.  Thus,  the  permutations  of  the  letters 
a,  b,  Cf  taken  two  at  a  time,  are 

ab,    ba,    aCf    ca,    bCy    cb. 
The  permutations  of  the  same  letters  taken  all  at  a  time,  are 

abCf    acb,    bac,    bca,    cab,    cba. 
It  is  evident  that  r  cannot  be  greater  than  n. 

464.  To  find  the  number  of  permutations  of  n  things, 
taken  r  at  a  time. 

Suppose  the  ti  things  to  be  n  letters,  a,b,  c,  d  .  .' ,  , 

The  number  of  permutations  of  n  letters,  taken  one  at  a  time, 
is  n. 

In  order  to  form  all  the  permutations  of  n  letters,  taken  two  at 
a  time,  we  must  annex  to  each  letter  each  of  the  n  —  1  other 
letters.     We  thus  obtain   n  {n  —  1)   permutations. 

In  order  to  form  all  the  permutations  of  n  letters,  taken  three 
at  a  time,  we  must  annex  to  each  of  the  permutations,  taken  two 
at  a  time,  each  of  the  n  —  ^  other  letters.  We  thus  obtain 
n(n  —  l)  {}i  —  2)  permutations. 


PERMUTATIOIS-S.  803 

In  the  same  manner  it  may  be  shown  that  the  number  of  per- 
mutations of  n  letters,  taken  4  at  a  time,  is  7i  (^^— 1)  (/^— 2)  (?^— 3). 

From  these  cases  it  might  be  conjectured  that  the  number  of 
permutations  of  n  letters,  taken  r  at  a  time,  is 

n{n  —  l){n  —  ^){n  —  d)  .  .  .  .  {n  —  r  +  1). 

To  show  that  this  is  true,  we  employ  the  method  of  mathemat- 
ical induction. 

Denote  the  number  of  permutations  of  n  letters,  taken  r  at  a 
time,  by  P^,  and  suppose  for  a  certain  value  of  r  that 

P,  =  ^  (7^  -  1)  (?i  -  2)  (;i  -  3)  .  .  .  .  (?^  -  r  +  1)  .    .    .     (A). 

Now,  in  order  to  form  all  the  permutations  of  n  letters,  taken 
r  -f  1  at  a  time,  we  must  annex  to  each  of  the  P^  permutations 
each  of  the  n  —  r  other  letters.    We  thus  form 

nin  —  l){n  —  2)(n  —  Z).,..  {n  —  r+l){n  —  r) 

permutations.    Hence,  denoting  the  number  of  permutations  of  n 
letters,  taken  r  +  1   at  a  time,  by  P^+i,  we  have 

Fr+i  =  n{n  —  l){n  —  2)(n  —  d)....  {n  —  r  +  1)  (n  —  r), 

which  may  be  written 

Fr^,  =  n{n-^l){n-2)(n-.3) {n-r  +  l)[n-{r^l)-\-l]. 

This  equation  is  of  the  same  form  as  (A) ;  that  is,  it  may  be 
derived  from  (A)  by  simply  substituting  r  +  1   for  r. 

If  then  (A)  is  true  when  r  is  a  certain  number,  it  is  true  when 
we  increase  that  number  by  one.  But  (A)  has  been  shown  to  be 
true  when  r  =  3 ;  it  is  therefore  true  when  r  ==  4 ;  it  is  there- 
fore true  when  r  =  5  ;  and  so  on.  Hence  the  formula  must  be 
universally  true. 

Cor. — K  r  =  n,   (A)  becomes 

'P^  =  n{n---l){n-2){n-^d)  .  .  ,  ^1    .    .    .    (B).    - 

That  is, 

T^ie  nnmher  of  permutations  of  n  things,  taken  n  at  a  time,  is 
equal  to  the  product  of  the  consecutive  numbers  from  1  to  n  in- 
clusive. 


304  PERMUTATIONS. 

For  brevity,  n(n —  1)  {n —  2)  {7i —  3)  .  .  .  .  1  is  often  de- 
noted by  the  symbol   \n,  which  is  resid,  factorial  n.  ^ 

465.  To  find  the  number  of  permutations  of  n 
things,  taken  n  at  a  time,  when  some  of  the  things  are 
identical. 

Suppose  the  n  things  to  be  w  letters ;  and  suppose  p  of  them 
to  be  a,  q  of  them  to  be  J,  r  of  them  to  be  c,  and  the  others  to  be 
uuhkc. 

Denote  the  required  number  of  permutations  by  N.  If  in  any 
one  of  these  N  permutations  thej3  letters  a  were  changed  into  p 
new  letters  different  from  each  other,  and  also  different  from  all 
the  other  letters  contained  in  the  permutation,  then,  without 
changing  the  situation  of  the  other  letters,  we  could  from  the 
single  permutation  form  [^  different  permutations;  therefore 
the  whole  number  of  permutations  would  be  N  x  [^  In  like 
manner,  if  the  q  letters  b  were  changed  into  q  new  lettei-s  differ- 
ent from  each  other,  and  also  different  from  all  the  other  letter's 
contained  in  the  N  x  \p  permutations,  we  could  form  N  x  |j'' 
X  1^  permutations;  and  if  the  r  letters  c  were  also  changed  \\\ 
the  same  way,  the  number  of  permutations  would  be  N  x  |/ 
X  [^  X  [r.  But  this  number  must  be  ecjual  to  the  number  d 
permutations  of  n  dissimilar  things,  taken  w  at  a  time;  hence. 

Nx[£x[£x[r  =  l£; 

\n 

whence,  ^  =  1 1= — r    •    •    •    (^« 

[£  X  [£  x|r  ^  ' 

466.  PB0BZEM8. 

1.  How  many  different  permutations  may  be  formed  of  8  let- 
ters, taken  5  at  a  time  ?  A7is.  6720. 

2.  How  many  different  permutations  may  be  formed  of  all  the 
letters  of  the  alphabet,  taken  4  at  a  time  ?  Ans.  358800. 

3.  How  many  different  permutations  may  be  made  of  6  things, 
taken  6  at  a  time  ?  Ans.  720. 


COMBIXATION-S.  305 

4.  now  many  different  numbers  may  be  formed  with  the  five 
figures,  5,  4,  3,  2,  1,  each  figure  occurring  once,  and  only  once,  in 
each  number?  Ans.  120. 

5.  How  many  different  permutations  may  be  made  of  the  letters 
in  the  word  Longitude,  taken  all  together  ?  Ans.  362880. 

6.  How  many  different  permutations  may  be  made  of  the  let- 
ters in  the  word  Caraccas,  taken  all  together?  A^is.  1120. 

7.  How  many  different  permutations  may  be  made  of  the  let- 
ters in  the  word  HeliopGlis,  taken  all  together  ?     Ans.  453G0O. 

8.  How  many  different  permutations  may  be  made  of  the  let- 
ters in  the  word  Ecclesiastical,  taken  all  together  ? 

^r^5.  .45-1053600. 

9.  What  value  must  7i  have  in  order  that  the  number  of  per- 
mutations of  n  things,  taken  4  at  a  time,  may  be  equal  to  12 
times  the  number  of  permutations  of  n  things,  taken  2  at  a  time  ? 

Ans.  n  =  0. 

COMBINATIONS. 

467.  T7ie  Combinations  of  n  things,  taken  r  at  a  time, 
are  the  results  obtained  by  arranging  the  things  in  as  many  differ- 
ent groups  of  r  each  as  possible,  without  regarding  the  order  in 
which  the  things  are  placed.  Thus,  the  combinations  of  the  let- 
ters a,  h,  c,  taken  two  at  a  time,  are 

ah,    ac,    he. 

It  will  be  observed  that  if  the  letters  be  regarded  as  factors, 
the  combinations  which  may  be  formed  by  taking  r  at  a  time 
will  constitute  all  the  different  products  of  the  r^^  degree,  of  which 
the  letters  are  capable. 

468.  To  find  the  number  of  combinations  of  n  things, 
taken  r  at  a  time. 

Denote  the  number  of  combinations  of  n  things,  taken  r  at  a 
time,  by  C^,  and  the  number  of  permutations  of  n  things,  taken  r 
at  a  time,  by  P,- 
20 


306  COMBINATIONS. 

It  is  evident  that  all  of  the  P^.  permutations  can  be  fonned  by 
gubjectiug  the  r  letters  of  each  of  the  C^  combinations  to  all  the 
per uuitat  10)18  of  which  these  letters  are  susceptible,  when  taken  r 
at  a  time.  Now,  the  number  of  permutations  of  r  letters,  taken 
r  at  a  time,  is  |r  (464,  Cob.)  ;  therefore  the  number  of  permuta- 
tions of  n  letters,  taken  r  at  a  time,  is  0^  [r ;  hence, 

C,  xlr  =  P,; 

p 

whence,  C,.  =  -p. 

But    P^=n(»-l)(»-2)(«-3)  ....  (w-r+1)  (464); 

_n(n-l)(;i-2)(;?-3)  .  .  .'.  (n-r+l) 
.  .    w-  ^  .    .    .     (1^}. 

469.  The  number  of  combinations  of  n  things,  taken  r  at  a 
time,  is  equal  to  the  number  of  combinations  of  n  things,  taken 
n  —  r  at  a  time. 

Denote  the  number  of  combinations  of  n  things,  taken  n  —  r 
at  a  time,  by  C„_r ;  then  by  (D), 

\,      _7i{n-l){n-2){n-3) [n-{n-r)  +  l] 


_n(n-l)(n-2)(7i-3) (r+l)  ,,, 

-  \n-r  '    '    '    ^^^' 

Multiplying  both  terms  of  the  second  member  of  (1)  by  |r, 

c^=tir|E:r  •  •  •  ('>• 

Multiplying  both  terms  of  the  second  member  of  (D)  by 


\n 
Gr  =  C^     .     .     .     (4). 


THE    BIXOMIAL    FORMULA. 


307 


470. 


PnOBLEMS. 


1.  Find  the  number  of  different  products  that  can  be  formed 
with  the  numbers  1,  2,  3,  4,  5,  taken  2  at  a  time.  Ans.  10. 

2.  What  vahie  must  n  have  in  order  that  the  number  of  per- 
mutations of  n  things,  taken  5  at  a  time,  may  be  equal  to  120  times 
the  number  of  combinations  of  n  things,  taken  3  at  a  time  ? 

Am.  8.     ^ 

3.  When  n  is  eren,  what  value  must  r  have,  in  order  that  Gr 
may  be  the  greatest  possible  ?  j  _  ^ 

Z 

4.  When  n  is  odd,  what  value  must  r  have,  in  order  that  C, 


may  be  the  greatest  possible  ? 


Ans.  r  z=z 


5.  From  a  company  of  soldiers  numbering  96  men  a  picket  of 
10  is  to  be  selected ;  in  how  many  ways  can  it  be  done  so  as  always 
to  include  a  particulai*  man  ?  |95 

6.  From  a  company  of  soldiers  numbering  96  men,  a  picket  of 
10  is  to  be  selected ;  in  how  many  ways  can  it  be  done  so  as  always 


to  exclude  a  particular  man  ? 


|95 
^"^^  [10xi85 


THE    BINOMIAL    FORMULA. 
x-\-  ah. 


471.  (x  +  a)(a;  +  ^=^  +  « 

{x  -\-  a)  {x  -\-  h)  {x  -\-  c)  =0?  -{-  a 

{x-\-a){X'\-b){x-\-c){X'\-d)=7^-\-a 


7?  ■\-  dbx-^-  abc. 
+  ac 
+  be 


a^-\-  ah 

a^-{-  abc 

+  ac 

■\-abd 

-\-ad 

+  acd 

4-  be 

-{-bed 

-\-bd 

-\-cd 

x  +  abcd. 


308  THE    BINOMIAL    FORMULA. 

In  each  of  these  identities  we  observe  the  following  laws : 

1.  The  number  of  terms  in  the  second  memher  is  one  greater 
than  the  number  of  binomial  factors  in  the  first  memher. 

2.  The  exponent  of  x  in  the  first  term  of  the  seco7id  member  is 
equal  to  the  number  of  binomial  factors,  and  in  each  of  the  suc- 
ceeding terms  the  exponent  of  x  is  one  less  than  in  the  preceding 
term. 

3.  Tlie  coefficient  of  the  first  term  of  the  second  member  is  unity ^ 
the  coefficient  of  the  second  term  is  the  sum  of  the  second  terms  of 
the  binomial  factors  j  the  coefficient  of  the  third  term  is  the  sum 
of  all  the  products  of  the  second  terms  of  the  binomial  factor s,  taken 
two  at  a  time  ;  the  coefficient  of  the  fourth  term,  is  the  sum  of  all 
the  products  of  the  second  terms  of  the  binomial  factors,  taken  three 
at  a  time  ;  and  so  on  :  the  last  term  is  the  product  of  all  the  second 
terms  of  the  binomial  factors. 

4!li2i»  That  the  laws  stated  in  the  preceding  Article  are  gen- 
eral may  be  shown  as  follows : 

Suppose  the  laws  to  be  true  in  the  case  of  n  binomials,  x  -f  a, 
X  -\-  by  X  -\-  c  .  .  .  .  X  -{-  k'y  that  is,  suppose 

{x  +  a)(x-{-b)(x-\-c)  ....  (a;-f /^•)=^"  +  Pi^""^  +  P2?^""HP3a^""^ 

+  Pn    .     .    .     (1), 

in  which    Pi  =  the  sum  of  the  terms  a,  b,  c  .  .  .  .  7c, 

Pg  =  the  sum  of  the  products  of  these  tei-ms,  taken  two 
at  a  time, 

Pg  =  the  sum  of  the  products  of  these  terms,  taken 
three  at  a  time, 


P„  =  the  product  of  all  these  terms. 
Multiplying  both  members  of  (1)  by  x  +  I, 
(x-^a){x  +  b){x+  c)  .  .  .  .  {x  +  Jc)  {x  +  l)  = 

a;«+i  -}-  Pj  a;"  +  Pg  3;^-^  +  P3   ^'"-^  ....  +PJ  .    .    •    (2). 


+  1 


+  Pi^ 


+   P2^ 


THE    BINOMIAL    FORMULA.  309 

Now  Pi+Z  =  a  +  Z>  +  c....+^'  +  ? 

=  the  sum  of  all  the  terms  a,  5,  c, .  .  .  .  Ic,  I; 

P2  +  Pi^  =  P3  +  (a  +  Z^  +  c -^k)! 

=  the  sum  of  the  products  of  all  the  terms 
a,b,c,....k,  I,  taken  two  at  a  time ; 

P3  +  Pg?  =  P3  +  {ab  +  ad  -\-  ac  +  he  ■\- hd  +  cd  .  .  .  .)l 

=  the  sum  of  the  products  of  all  the  terms 
a,  bf  c,  .  .  .  .  h  If  taken  three  at  a  time . 


PJ  =  the  product  of  all  the  terms  a^b,  c, . . . .  k,  I 

The  law  of  the  exponents  in  (2)  is  also  the  same  as  in  (1). 

Hence,  if  the  laws  are  true  when  n  factors  are  used,  they  will 
be  true  when  7^  -f  1  factors  are  used.  But  they  have  been  shown 
to  be  true  when  ?j  =  4;  therefore  they  are  true  when  oi=:b; 
and  so  on.     Hence  the  laws  must  be  true  universally. 

473.  The  number  of  terms  in  P^  is  obviously  n;  the  number 
of  terms  in  P2  is  equal  to  the  number  of  combinations  of  n  thiugs, 

41  ( ji  -_  1  ^ 
taken  two  at  a  time,  that  is,  — ^j^ — -  (468) ;   the  number  of 

terms  in  P3  is  equal  to  the  number  of  combinations  of  n  things, 

taken  three  at  a  time,  that  is,   — ^^ — ;    and  so  on. 

Now  suppose       a=:b  =  c=.,..k;       then       Pj  =  na, 
_n{n  —  l)^  p    _  n  {n  —  l){n—  2) 

Under  this  hypothesis,  (1)  of  Art.  473  becomes 

(X  +  a)-=x-  +  7iaa^-'  +  n{n-l)  ^^^^_^  _^  n{n-l){n-2)  ^^^^_^  ^ 

If 
This  is  the  Binomial  Formula,    The  second  member 
of  this  formula  is  called  the  Expansion  or  Developtnent 

of  {x  +  a)^,  and  when  we  put  this  expansion  or  development  in 
the  place  of    {x  +  a)^  we  are  said  to  expand  or  develop  (x  -f-  a)". 


P^  =  ^^Z^ a2,         P3  ^  'lK!i_tJS-^±l „s,       and  so  on. 


310  THE    BINOMIAL    FORMULA. 

474.  The  coefficient  of  the  product  of  the  powers  of  a  and  x 
in  any  term  of  the  expansion  of  {x  +  a)«  is  called  the  coefficient 
of  that  term.    Thus,  the  coefficient  of  the  thu'd  term  of  the 

«  /          N     •     ^  (^  —  1) 
expansion  of  (x  +  aY  is   tt . 

475.  The  first  letter  in  an  expression  of  the  form  of  {x  +  «)** 
is  called  the  leading  letter, 

476.  Another  Proof  of  the  Binomial  Formula,— 

We  can  verify  the  Binomial  Formula  by  trial  for  small  values  of 
^  as  2,  3,  4. 
Assume 

If  l£ 

+ +«"    .    .    .    (1). 

Multiplying  both  members  of  (1)  hj  x  ■\-  a  and  reducing, 

+^fc^a»."-H  .  .  .  .  +«»-    .   .   .   (^); 

that  is,  the  expansion  of  {x  +  fl)"+*  is  of  the  same  form  as  that 
of  {x  H-  a)".  Hence,  if  the  Binomial  Formula  is  true  for  any  ex- 
ponent, it  is  true  when  the  exponent  is  increased  by  unity.  But 
the  formula  is  true  when  w  =  4;  it  is  therefore  true  when  w=5; 
it  is  therefore  true  when  w  =  6 ;  and  so  on.  Hence  the  Bino- 
mial Fonnula  is  true  for  any  positive  integral  exponent. 

Cor.  1. — If  we  multiply  the  coefficient  of  any  term  in  the  ex- 
pansion of  {x  +  aY  hy  the  exponent  of  x  in  that  term,  and  di- 
vide the  'product  hy  the  number  obtained  by  adding  1  to  the  expo- 
nent of  a  in  the  same  term,  the  quotient  will  be  the  coefficient  of 
the  succeeding  term. 

Cor.  2. — Tlie  sum  of  the  exponents  of  a  and  x  in  any  term  of 
the  expansion  of  {x  +  aY  is  equal  to  n. 


THE    BINOMIAL    FORMULA.  311 

477.  To  find  the  stun  of  the  coefficients  in  the  ex- 
pansion of  {x  4-  a)\ 


The  formula 


(J^l  .2.n-2 j_ nin-l){n-2)   .  _ 


{x  4-  ay = a;"  +  naaf"-^  -\ — ^-uT--  «^^"~^  +  -^ ^ d^^ 

+  .  .  .  .   H-a» 

is  true  for  all  values  of  x  and  «,  and  the  coefficients  are  indepen- 
dent of  x  and  a.    Suppose  a;  =  c  =  1 ;  we  then  have 

n{n  —  V)       n(n  —  1)  (7i  —  2)    .  ,   ^ 

2»=  1  +  w  +  -^^ — ^  +  -^ ^ + +1. 

That  is,  the  sum  of  the  coeflScients  in  the  expansion  of  {x-\-aY 
is2». 

478.  To  find  the  r^^  term  of  the  expansion  of 
(x  4-  ay. 

The  exponent  of  a  in  any  term  of  the  expansion  of  {x  +  aY  is 
one  less  than  the  number  of  that  term ;  hence  the  exponent  of  a 
in  the  r^  tenn  is  r  —  1.  The  sum  of  the  exponents  of  a  and  x 
in  any  term  is  n ;  hence  the  exponent  of  x  in  the  r^^  term  is 
n  —  r-\-l.  The  coefficient  of  the  second  term  is  equal  to  the 
number  of  combinations  of  n  things,  taken  one  at  a  time ;  the  co- 
efficient of  the  third  tenn  is  equal  to  the  number  of  combinations 
of  n  things,  taken  Uvo  at  a  time  ;  and  so  on  ;  hence  the  coefficient 
of  the  r'*  term  is  equal  to  the  number  of  combinations  of  n  things 
taken  r  —  1   at  a  time ;  that  is, 

n(n-l){n-%){n-^).  .  ,  ,  (^  -  r  +  2)  ^^^^^^ 
|r  —  1 

Therefore,  denoting  the  r^  term  by  T„  we  have 

*■  |r  — 1 

It  should  be  observed  that  r  cannot  be  greater  than  w  -|- 1. 


312  THE    BIN-QMIAL    FORMULA. 

479.  In  ilic  expansion  of  (x  +  «)'»  Me  coeficicnf  of  the  r^^ 
term  from  the  heyinning  is  equal  to  the  coefficient  of  the  r^  term 
from  the  end. 

The  coefficient  of  the  r^  term  from  the  beginning  is 

n(n-l){n^2){n-^3)  ,   ,   ,  .  (n  -  r -i- 2)  ^^^^^^ 

Multiplying  both  numerator  and  denominator  by  \n  —  r  4  1, 
this  becomes 

|r  —  1  X  \n  —  r-i-  1' 

The  r^  tenn  from  the  end  is  the  {71  —  r  -{-  2y^  term  from 
the  beginning,  and  its  coefficient  is 

n{n-'i){n-2)(n-3) [n-(n-r  +  2)-{-2] 

\n  —  r  -\-  1  ^        '* 

which  becomes  by  reduction 

n{n^l){n—2){n—Z)  .  .  .  .  r 

N-r  +  l 

Multiplying  both  numerator  and  denominator  by  \r  —  1,  this 
becomes 


\r  —  1  X  \n  —  r-\-  1' 


480.  To  obtain  the  Expansion  of  (x  —  «)»,  it  is  suffi- 
cient to  put  —  a  in  the  place  of  -\-  a  in  the  expansion  of 
(x  +  a)«  The  tenns  which  contain  the  odd  powers  of  —  a  will 
be  negative,  and  the  terms  which  contain  the  even  powers  of  —  a 
will  be  positive.     Hence, 

If  E 

,  "(«-i)(«-2)f'i:z^^4^„-4 .  _  . 


^ 


a\    a^,    a\ 

a\    a\    cfi; 

V,    b\    V, 

b%     ¥,    ¥; 

1,    5,   10, 

10,     5,     1  (4'76,CoR.l), 

THE    BINOMIAL    FORMULA,  313 

481.  EXAMPZES. 

1.  Expand   (a  4-  ^)^- 

First  Solution. — In  the  expansion  of  {a+by  the  powers  of  a  are 

the  powers  of  b  are 
and  the  coeflBcients  are 

(a  +  Z>)5  =  flS  ^_  5^4^  ^  i0a3^2  _f_  10^2^3  +  5^^  _|_  js. 

Second  Solution. — The  Hteral  parts  of  the  terms  of  the  expan- 
sion of  (a  +  by  are 

a^    a%    aW,    aW,    a¥,    b\ 
and  the  coefficients  are       1,     5,      10,      10,       5,     1 ; 

(a  H-  5)5  =  ^5  +  ba'^b  +  10^3^^  +  10^2^,3  4.  ^a¥  +  5^ 

Third  Solution. — Substituting  a  for  x,  b  for  a,  and  5  for  n,  in 
the  Binomial  Formula,  we  have 

{a-\-by  =  a^  +  5a*b  +  lOaW  +  lOa^h^  +  6aM  +  b^. 

2.  Expand   {2x  —  Say. 

Powers  of  22-,  16a:*,        Sa^,    Aa^,  2x,    (2a:)0; 

Powers  of  —da,     (—  Say,     —  Sa,    9a\     —  27a^,      81a*; 
Coefficients,  1,  4,       6,  4,  1; 

.-.      (2x  —  Say  =  Ux^  —  ^^ax?  +  216«2|;2  _  2l6aSa;  +  81a*. 

3.  Expand   (a -\- b  -^  c -\- dy. 

Put  ic  =  a  +  i   and  y=zc-^d\   then 
(fl  +  5  +  c  +  t?)3  =  (a;  +  «/)3  =  a;3  ^  3^4}^  _f_  33.^2  ^  ^^s 

Substituting  for  x  and  y  their  vakies, 

(a  +  &  +  c+(^)3=(a4-5)3  +  3(fl  +  ^')2(c4-6?)+3(a4-5)(c  +  6?)2-f(c  +  c?)3 
=a3  ^  30^2^  ^  3^j2  +  53  ^  3^2^  _}.  6ar^,c  +  3^2^  +  3^2^ 
+  ^abd  +  SbH  +  3ac2  +  Gac^+3a6?2  4-35c2+65crf 
+  3i(^  +  c3  +  3c2(/  +  Scd'^  +  <^. 


814  THE    BINOMIAL    FORMULA. 

4.  Find  the  5th  term  of  the  expansion  of  {a  +  ly^. 
Substituting  a  for  x,  b  for  a,  15  for  ??,  and  5  for  r  in  the 

formula  of  Art.  4T8,  we  have 

^        15  X  14  X  13  X  12  y.  ..       -ion-ii.  11 
^  1x2x3x4 

5.  Expand   {a  —  hf. 

Ans.  flS  —  5^45  ^  X0a3i2  _  i^^aW  ^  5^j4  _  js. 

6.  Expand   (1  +  cf. 

Ans,  1  +  6c  +  15c2  +  20^8  +  15c*  +  Sc^  +  c«. 

7.  Expand   {x  +  y)". 

A  718.  x'  +  7a:«?/  +  21x^if + 35ar*j^3  ^  35^:8^ + 212^2^8 + 73:^6  ^  ^7, 

8.  Expand   (a^  _  1)8. 

Ans.  ai«-8fliH28ai2_56aio+70a8-56a6  +  28fl4-8a2  +  l. 

9.  Expand   (a  —  c)^. 

—  36flV  +  9ac8  —  c*. 

10.  Expand   (1  -\-axy. 

Ans.  1  +  5fl«;  +  lOa^x^  +  lOa^s^  4-  50*^  +  a^^^ 

11.  Expand   (3«-f-2c)^ 

A ns.  243^5 + SlOa^c  + 1080^3^2  +  720^^  +  240ac* + 32^5. 


12.  Expand   (o-l)^. 


>!       -.rc^o^     010-     .3125„     625-125,      5^',       x^ 
Ans.  15620-3120.^  +  -^:^-— ^+^3^^-*-^  +  -4^. 

13.  Expand   (^2  —  ah  -\-  lf)K 

Ans.  «» —  ^a?h  +  lOa^Z^  —  Ua^h^  +  19a*Z»*  —  IGa^Z^s  4.  i^a^^ 
—  4m1P  +  ¥. 

14.  Find  the  middle  term  of  the  expansion  of  {a  +  xy^. 

Ans.  '2l2a^x\ 

(3a      4r\* 

16.  Find  the  2001»'  term  of  the  expansion  of   («^  +  x'^J* 

Ans.  2003001a^^a;«». 


THE    'nP^    KOOT    OF    QUANTITIES.  315 

THE    n^^    ROOT    OF    QUANTITIES. 
433.  To  find  the  n^^  root  of  a  polynomial. 

Find  the  vJ'^  root  ofa;"  +  y^Q^"~^  +  ^^  ^^,1"     aH''-'^  ....   +  a". 

if 

x^  +  waa^-i  4-  ^1.7"      Q^^^""^  .  .  .  .   +  a'^Iic  +  a 

if 

Arrange  the  terms  according  to  the  descending  powers  of  x. 
The  n^^^  root  of  the  first  term,  a;",  is  x,  which  is  the  first  term  of  the 
required  root.  The  second  term  of  the  root  may  be  found  by 
dividing  the  second  term  of  the  given  polynomial  by  wa;«~^ 

If  there  were  more  terms  in  the  root,  we  should  proceed  with 
X  -\-  a  as  we  did  with  x, 

RULE. 

I.  Arrajige  the  given  polynomial  according  to  the  powers  of  one 
of  its  letters. 

II.  Extract  the  n^^  root  of  the  first  term;  the  result  will  be  the 
first  term  of  the  required  root.  Subtract  the  n^  power  of  this 
term  from  the  given  polynomial. 

III.  Divide  the  first  term  of  the  remainder  by  n  times  the 
(n  —  1)^  poiver  of  the  first  term  of  the  root  ;  the  quotient  will  be 
the  second  term  of  the  root.  Subtract  the  n^^  power  of  the  sum  of 
the  first  and  second  terms  of  the  root  from  the  given  polynomial. 

IV.  Take  n  times  the  {n  — 1)<*  potver  of  the  sum  of  the  first 
and  second  terms  of  the  root  for  a  second  divisor.  Divide  the  first 
term  of  the  second  remainder  by  the  first  term  of  the  second  divisor  ; 
the  quotient  will  be  the  third  term  of  the  root.  Subtract  the  n^^ 
power  of  the  sum  of  the  first,  second,  and  third  terms  of  the  root 
from  the  given  polynomial. 

V.  Proceed  in  this  manner  until  all  the  terms  of  the  root  have 
been  found. 


316  THE    W'^    ROOT    OF    QUANTITIES. 

Con. — If  the  root  contains  only  two  terms,  it  may  be  obtained 
by  extracting  the  n^  root  of  the  extreme  terms  of  the  arranged 
polynomial,  and  placing  the  proper  sign  between  the  results. 
Thus,  the  cube  root  of  a^  ^-  Za^h  +  oaW'  ■\-'b^  is  a-^l,  and  the 
cube  root  of  a^  —  Zd^h  +  ZaU^  —  W  is  a  —  h. 

EXAMPLES. 

1.  Find  the  fourth  root  of  16a*  —  96a^  H- 216fl2ic3  +  Sla;^ 
—  216aa:8,  j^^g^  2a  —  Zx. 

2.  Find  the  fifth  root  of  SOa^  +  32^5  _  SOa*  —  40^2  +  10a— 1. 

Ans.  2a  —  1. 

3.  Find  the  fourth  root  of  336^5  +  Sia^  —  21Ga^  —  56a*  +  16 
— 224a3 + 64a.  A  ns,  3a^  —  2a  —  2. 

4.  Find  the  fourth  root  of  a*  —  ^^b  +  Qa?b^  —  4aJ3  -|-  l\ 

Ans.  a  —  h, 

5.  Find  the  fifth  root  of  cfi+ba^h  +  lOaW+10aW-\-oa¥-\-l)\ 

Ans.  a  +  h. 

6.  Find  the  sixth  root  of  a^— 6a5  4-15a*— ^Oa^  +  lSa^— 6a  +  l. 

Ans.  a  —  1. 

7.  Find  the  seventh  root  of  a7+7a«+21a«+35a*+35a3+21a2 
+  7a  +  l.  Ans.  a  -\-\. 

8.  Find  the  eighth  root  of  3?—%x'^-\-2^ofi—hQj^+l^x^-bQ7? 
+  28arJ— 8a;  +  l.  Ans.  x  —  1. 

483.  To  find  the  71^^  root  of  a  number. 

For  a  reason  similar  to  that  given  in  Art.  271,  we  separate 
the  given  number  into  periods  of  71  figures  each,  beginning  with 
units.  The  71^  root  of  the  greatest  n^^  power  contained  in  the 
period  on  the  left  will  be  the  first  figure  of  the  root.  If  we  sub- 
tract the  71^^  power  of  the  first  figure  from  the  given  number,  and 
divide  the  remainder  by  n  times  the  (71  —  1)^^  power  of  that 
figure,  regarding  its  local  value,  the  quotient  will  be  the  second 
figure  of  the  root,  or  a  figure  too  large.  The  result  may  be  tested 
by  subtracting  the  71^^  power  of  the  number  represented  by  the 


SYNOPSIS    FOR    EEVIEW. 


317 


first  and  second  figures  of  the  root  from  the  given  number.  If 
there  are  additional  figures  in  the  root,  they  may  be  found  in  the 
same  manner. 

:exampIjJes. 

1.  Find  the  fifth  root  of  33554432. 

335,54432(30  +  2  =  32 
305  _  243  00000 

5  X  304  =  4050000)  92  54432 
325  ^  33554432 

2.  Find  the  fourth  root  of  79502005521.  A7is.  531. 

3.  Find  the  fourth  root  of  75450765.3376.  Ans.  93.2. 

4.  Find  the  fourth  root  of  2526.88187761.  Ans.  7.09. 

5.  Find  the  fifth  root  of  418227202051.  Ans.  211. 

6.  Find  the  seventh  root  of  34359738368.  Ans.  32. 


484. 


SYNOPSIS    FOR    REVIEW. 


PEEMUTATIONS 


OOMBINATIONS, 


BINOMIAL  FOKMULA. 


EXTRACTION  OP 
HIGHEE.  KOOTS. 


n  THINGS  TAKEN  r  AT  A  TIME. 
n  THINGS  TAKEN  71  AT  A  TIME. 
n  THINGS  TAKEN  71  AT    A  TIME,    WHEN 

SOME  ARE  IDENTICAL. 
7?    THINGS  TAKEN   T  AT  A  TIME. 

Interpret  the  equation  C^  =  C„--r. 

Product  of  n  binomials  whose  first 
terms  are  identical  and  whose 
second  terms  are  different. 

General  laws.     1,  2,  3. 

Binomial  Formula. 

Coefficient  of  a  term  of  the  ex- 
pansion OP  (a:  +  a)". 

Leading  letter. 

Another  proof  op  Binomial  For- 
mula.   Cor.  1,  2. 

To  find  the  sum  op  coefficients  in 

THE  expansion   OF   {x  +  O,)^. 

To  FIND  THE  r'*  TERM  OF  THE  EXPAN- 
SION OF  {x  -\-  ay. 

Expansion  of  {x — «)". 

To  FIND  THE  n'^  root    OF    A    POLYNO- 
MIAL.    Rule.     Cor. 
To  FIND  THE  7l'*  ROOT  OF   A   NUMBER. 


CHAPTEE   XIX. 
IDEIfTIOAL     EQUATIOI^fS, 


PROPERTIES  OF  IDENTICAL  EQUATIONS. 
485.  Jf  the  equation 
A  +  B.r  +  Cu^5  +  D^+  etc.  =  A'  +  B'a;  +  C'a;2+D'^H  etc., 

in  tuliich  A,  B,  C,  D,  etc.,  A',  B',  C,  D',  etc.,  are  finite  quantities 
independent  of  x,  is  an  identity y  the  coefficients  of  the  like  powers 
of  X  are  equal  to  each  other. 

Since  this  equation  ig  tnie  for  every  value  that  may  be  assigned 
to  X  (178),  it  must  be  true  when  a;  =:  0.  But  when  x  =  0,  all 
the  terms  disappear  except  A  and  A',  and  the  equation  becomes 

A  =  A'. 

Dropping  A  from  one  member  of  the  given  equation,  and  A' 
from  the  other, 

B^;  +  Ca;'2  +  Bx^  +  etc.  =  B'x  -f-  C'x^  +  B'x^  +  etc. 

Dividing  both  members  of  this  equation  by  a:, 

B  -{•  Cx  -{-  Da;2  -\-  etc.  =  B'  +  C'x  +  D'x^  +  etc 

Making  a;  =  0,  as  before,  this  equation  becomes 

B  =  B'. 

In  like  manner  it  may  be  shown  that 

C  =  C', 

D  =  D',  etc. 


DECOMPOSITION    OF    RATIONAL    FEACTIONS.  319 

486.  If  the  equation 

A  +  Bir  +  Ca;2  +  Da:3  _^  etc.  =  0 

15  an  identity^  each  of  the  coefficients  A,  B,  0,  D^  etc.,  is  equal  to 
zero. 

Since  this  equation  is  true  for  every  value  that  may  be  assigned 
to  X,  it  must  be  true  when  a;  =  0.  But  when  a;  =  0,  the  given 
equation  becomes 

A  =  0. 

Dropping  A  from  the  given  equation,  and  dividing  the  rqsult 

by  a;, 

B  +  Ca;  +  D:c2  _|.  etc.  =  0. 

Making  a;  =  0,  as  before,  this  equation  becomes 

B  =  0. 
In  like  manner  it  may  be  shown  that  • 

C  =  0, 
D  =  0,  etc. 

487.  Undetermined  Coefficients  are  such  as  are  un- 
known in  an  assumed  identity.  Thus,  if  we  assume  {x  +  aY  = 
Ax'^  +  Ba:2  j^  Qrf.  _i_  J)  to  be  identically  true.  A,  B,  C,  and  D  are 
undetermined  coefficients. 

DECOMPOSITION  OF  RATIONAL  FRACTIONS. 

488.  To  Decomimse  a  Mat  ion  al  Fraction  is  to 

separate  it  into  fractions  whose  sum  is  equal  to  the  given  fraction 
and  the  product  of  whose  denominators  is  equal  to  the  given  de- 
nominator. The  parts  into  which  the  given  fraction  is  separated 
are  called  Partial  Fractions. 

EXAMPZES. 

1.  Separate  -^ — -r   into  partial  fractions. 

X  ~—  iX  ~p  xo 

The  factors  of  the  denominator  are  x  —  b  and  rr  —  2 :  hence 


320  IDENTICAL    EQUATIONS. 

the  denominator  of  one  of  the  partial  fractions  is  x  —  6,  and 
that  of  the  other  is  a;  —  2. 

.  8a: -31  A       .       B  ... 

Assume  -r— -, — —r?.  = ^  H :t    •    •    •     (I)' 

2^2  _  7a:  +  10       a;  —  5       a:  —  2  ^  ^ 

Since  the  first  member  is  the  sum  of  the  two  fractions  in  the 
second  member,  this  equation  is  an  identity. 

Clearing  (1)  of  fractions  and  uniting  similar  terms, 

8a:-31^(A  +  B)a;-(2A  +  5B)     .    .    .     (2). 

(2A4-5B=:31)    ^        ^' 
whence,  A  =  3    and    B  =  5. 

Substituting  3  for  A  and  5  for  B  in  (1), 

8a:  — 31  3  5 


a:*  —  7a:  +  10       x  —  5       a:  —  2 

72^  4-  X 

2.  {Separate  -. -r-rr -^  into  partial  fractions. 

^  (ar-|.l)(2ar  — 1) 

.  W  +  x  A      ,       B  .^. 

Assume  ^^  ^  ^^  ^^^  _  ^^  ^ —^  +  ^— -^    ...    .    (1). 

Clearing  of  fractions  and  uniting  similar  terms, 

7ar»  +  a:  =  (2A  +  B)  a:  +  B  —  A    .     .    .     (2). 

The  coefficient  of  a^  in  the  second  member  of  (2)  is  0 ; 

7  =  0     (485), 

which  is  absurd.     Hence  the  given  fraction  cannot  be  separatee! 
into  partial  fractions  having  numerators  independent  of  x. 

.  7a;2-|-a:  Aa:      ,       Ba:  ,. 

Assume   -. —-rz -r  = +  7i t    •    •    •     {pJ- 

(a;  +  1)  (2a:  —  1)       a:  +  1       2a:  —  1  ^  ^ 

Clearing  of  fractions  and  uniting  similar  terms, 

7a?J-ha:=(2A  +  B)a:2+(B-A)a:    .     .    .     (4). 


SYNOPSIS    FOR    REVIEW.  321 

Equating  the  coefficients  of  like  powers  of  x  in  (4), 

(  2A  +  B  =  7  ) 

(      B-A=:lP 

whence,  A  =  2    and    B  =  3. 

Substituting  2  for  A  and  3  for  B  in  (3), 

72;2  +  a;  2^-         _  Zx 


{x  4-  1)  (2a;  —  1)       x^\   '    'Zx—\ 

From  this  example  we  learn  that  if  we  assume  an  impossible 
form  for  the  partial  fractions,  the  fact  will  be  made  apparent  by 
some  absurdity  in  the  equations  of  condition  (179). 

Separate  each  of  the  following  fractions  into  its  partial  fractions : 
7a; -24  5,2 


20^:4-2  .  8  6 

2.6-2-1- 3a;  — 20*  ^^*  2a;  —  5  "^  a;  +  4* 

6arJ-22a;  +  18  .  1       ,       2       ,       3 


(a;— l)(a;2  — 5a;  + 6)*  'a;- l^a;  — 2   '   .^•  — 3' 

1  3 

^     a;  +2  .2,22 

6.  ,        .  Ans,  — — r  H r . 

a;  -[-  1       a;  —  la; 

1111 

.2  2  3^3 

7.  —, T^-^ rrrr.     Arts.  — -— —  - — — r  + 


y?- 

-x' 

10 

a4  _  xZx^  ^  36 
3:^3  -}-  5a;2  —  2a; 

a;-}- 2       X  —  2       a;-|-3       a;  —  3 
8.    ^ -^ .  An8. + 


a;2  —  1  a;  —  l'a;  +  l 

489.  SYNOPSIS    FOR    REVIEW. 


t— (  n 

H  EH 
w  p 

fi  ex 


Properties  of  Identical  Equations.  \        * 

Undetermined  Coefficients. 

Decomposition  of  Rational  Fractions.  Partial  Fractions. 
21 


CHAPTER   XX. 
SERIES. 

GENERAL    DEFINITIONS. 

490,  A  Series  is  a  succession  of  quantities,  each  of  which, 
except  the  first,  or  a  certain  number  of  the  first,  may  be  derived 
from  the  preceding  one,  or  a  certain  number  of  the  preceding 
ones, by  a  fixed  law  called  the  Law  of  the  Series,    Thus, 

1,     2,     3,     4,     6,     6,     7,     8,     9, 

is  a  series,  the  law  of  which  is  that  each  quantity,  except  the  first, 
is  derived  from  the  preceding  one  by  adding  unity  to  it. 

491,  Tlie  Terms  of  a  series  are  the  quantities  of  which  the 
series  consists. 

492.  A  Finite  Series  is  one  which,  by  its  law  of  forma- 
tion, can  have  only  Q.Ji7iUe  number  of  terms.  Such  a  series  is  said 
to  terminate.  Thus,  the  expansion  of  {x  -f  a)»,  when  n  is  a  pos- 
itive integer,  is  a  finite  scries. 

493.  An  Infinite  Series  is  one  which,  by  its  law  of 
formation,  may  have  an  infinite  number  of  tenns.  Such  a  series 
is  said  not  to  terminate.     Thus, 

1     1     1     -L     _L     JL 
*     2'    4'    8'     16'    32'  .  64' 


is  an  infinite  series. 

494.  A  Converging  Series  is  an  infinite  series,  the  sum 
of  the  first  n  terms  of  which  cannot  numerically  exceed  some  finite 
quantity,  however  great  n  may  be.    Thus, 


ARITHMETICAL    PROGRESSION.  32S 


1       1       1       J_ 

^'     2'     4'     8'     !♦>' 


is  a  converging  series,  for  the  sum  of  its  first  n  terms  cannot 
exceed  2,  however  great  n  may  be. 

495.  A  Divevffing  Series  is  an  infinite  series,  the  sum  of 
the  first  n  terms  of  which  can  be  made  numerically  greater  tlian 
any  finite  quantity  by  taking  n  sufiiciently  great.    Thus, 


1,    2,    3,    4,     5,     6,     7,     8, 
is  a  diverging  series. 


ARITHMETICAL    PROGRESSION. 

496.  An  Arithinetical  JProgression ,  or  a  PrO" 
ffvession  by  Ulfference,  is  a  series  in  which  the  difference 
between  the  first  and  second  terms  is  equal  to  the  difference 
between  any  other  two  consecutive  terms.  Thus,  1,  3,  5,  7,  9  is 
an  arithmetical  progression. 

An  arithmetical  progression  is  sometimes  called  an  Avith- 
metical  Series. 

For  brevity  we  shall  sometimes  use  A.  P.  for  the  phrase 
arithmetical  progressio7i. 

497.  The  Ertrenies  of  an  A.  P.  are  the  first  term  and  the 
last  term  ;  the  other  terms  are  the  Means, 

498.  The   Common  Difference  of  an  A.  P.  is  the 

remainder  obtained  by  subtracting  any  term  from  the  one  which 
follows  it.  Thus,  in  the  progression  1,  3,  5,  7,  9,  the  common 
difference  is  2. 

499.  An  Increasinf/  A,  JP.  is  one  in  which  the  com- 
mon difference  is  positive.  Thus,  1,  2,  3,  4^  5  is  an  increasing 
A.  P. 

500.  A  Decreasing  A.  P.'i^  one  in  which  the  common 
difference  is  negative.     Thus,  9,  7,  5,  3,  1  is  a  decreasing  A.  P. 


324  SERIES.  , 

501.  Notation. — In  treating  arithmetical  progressions  we  shall 
use  the  following  notation : 

a  =  the  first  term  of  the  progression, 
I  =  the  last  or  n^^  term, 
d  =  the  common  difierence, 
w  ^  the  number  of  terms, 
s  =  the  sum  of  all  the  terms. 

Thus,  in  the  A.  P.         1,  3,  5,   7, .  9, 

a  — I,   1  =  9,  d  =  2,  w  =  5,   s  =  25. 

502.  To  find  I  when  a,  d,  and  n  are  given. 

The  first  term  is  «,  the  second  term  is  a  +  d,  the  third  term  is 
a  +  2eZ,  the  fourth  term  is  a  +  M,  and  so  on;  hence  the  n^ 
term  is  a  ■}-  {n  —  1)  d;  that  is, 

l  =  a  -\-  (n  —  1)^. 

503.  To  find  s  when  a,  I,  and  n  are  given. 

sz=za-\-{a-^d)-\-{a  +  2d)+{a-\-M)+ .  .  .  .  +1    .     .     .     (1). 
Inverting  the  order  of  terms  in  the  second  member  of  (1), 

s  =  l^{l-d)  +  {l—2d)  +  {l^Zd)-\-,..,-\-a    .     .     .     (2). 
Adding  (1)  and  (2), 
25=(rt  +  0  +  («  +  0  +  («  +  0+---- +(«  +  0=^(«^  +  0  .  .  .  (3); 

whence,  « =  o  (^  +  0     •    •    •     (^)- 

504.  In  an  A.  P.  the  sum  of  any  ttvo  terms  equidistant  from 
the  extremes  is  equal  to  the  sum  of  the  extremes. 

Let  X  denote  a  term  which  has  m  terms  before  it,  and  y  a  term 
which  has  m  terms  after  it ;  then 

-whence,  x  -\-  y  =  a  -\- 1 


AKITHMETICAL    PROGRESSIOIT.  325 

505.  To  insert  any  number  of  arithmetical  means 
between  two  given  quantities. 

Let  a  and  b  be  the  given  quantities,  and  let  it  be  required  to 
insert  m  arithmetical  means  between  them;  that  is,  let  it  be 
required  to  form  an  A.  P.  whose  extremes  are  a  and  b  and  the 
number  of  whose  terms  is  w  -{-  2. 

Substituting  b  for  I  and  m  +  2  for  n  in  the  formula  of 
Art.  503,  we  have 

b  =  a  +  {m  -{-l)d; 
b  —  a 


whence,  d  = 


m+  1 


By  adding  the  common  difference  to  a  we  obtain  the  second 
term ;  by  adding  it  to  the  second  term  we  obtain  the  third ;  and 
so  on. 

Example. — ^Insert  10  arithmetical  means  between  5  and  38. 
38-5_ 

hence  the  required  progression  is 

6,  8,   11,   14,   17,   20,  23,  26,   29,  32,  35,  38. 

506.  To  find  any  two  of  the  quantities  a,  Z,  J,  n, 
and  5,  when  the  three  others  are  given. 

ilz=a-\-{n-l)d\ 
The  group  j,=  |(«  +  o  [ 

contains  the  five  quantities  a,  I,  d,  n,  and  s ;  hence  any  two  of 
them  may  be  found  when  the  three  others  are  given. 

The  ten  cases  are  given  in  the  following  table  as  an  exercise 
for  the  student. 

Each  case  is  an  example  of  two  simultaneous  equations  with 
two  unknown  quantities. 


326 


SERIES. 


OITEN. 

TO  FIND. 

a,  d,  n 

ly         S 

a,  dy  I 

W,    5 

a,  d,  s 

Hy       I 

rt,  w,   I 

d,  S 

a,  71,  s 

dy       I 

a,  I,  s 

dy  n 

d,  n,  I 

a,  8 

dy     fly     8 

ay    I 

d,  h  s 

a,  n 

I,       fly      S 

ayd 

BESULTING  FOKMDUB. 


d—%a±  V(2a  —  d)'^  +  8^6- 
2d 


10. 


J        I  —  a  W/      .    n 

d  =  ^--^,    l^^-^-a. 
n  (n  —  1)  n 

.  ^  —  a^  28 

d  =  -; :,       71  = 1. 

2s  —  a  —  l  a  -i-  I 

a=l-{7i^l)dy    s=^[2I-(7i-l)d], 

_2s—7i{n—\)d   j_2s-\-n{7i—l)d 
""-  271  '  ^-  2n  • 

^-i{d±  V(zl  +  df  -  iidsjy 
_2l-\-d±  V(2l  +  d)^-  Sds 
2d 
2s  2  (In  —  s) 

n  71  {n  —  1) 


n  = 


507. 


PJtOBLEMS, 


1.  The  first  term  of  an  A.  P.  is  5,  the  common  difference  is  3, 
and  the  number  of  terms  is  24.  Find  the  last  term  and  the  sum 
of  all  the  terms. 

AVe  have  given  a  =  5,  c?  =  3,  n  =  24 ; 

?  =  5  + (24-1)3  =  74, 


and 


24 


s  =  y  [10  +  (24  -1)3]  =  948    (506,  1). 


ARITHMETICAL    PROGRESSION.  32? 

After  finding  the  value  of  Z,  we  might  have  found  the  value  of 
s  from  the  formula    s  =  -  (a  +  /).    Thus, 

04. 

s  =  ^  (5  +  74)  =  948. 

2.  Given  «  =  15,  tZ  =  —  2,  and  5  =  60,  to  find  I  and  n, 

?  =  -  ^-  ±  1/2  (-  2)  60  +  (15  -  ^-/  =  5   or   -3, 
and 

Both  values  of  n  are  possible ;  for  there  are  two  progressions 
which  satisfy  the  given  conditions,  one  having  6  terms,  the  other 
10 ;  these  progressions  are 

15,    13,    11,     9,     7,    5, 
and         15,    13,    11,    9,    7,    5,    3,     1,     —1,     —3. 

Another  Solution, — Substituting  the  given  values  in  the  group 
ilz=za  +  {n  —  l)cl\ 

we  obtain  the  group 

i     Z=15-2(7^-l)  ^ 
J60  =  |(15  +  0  )' 

whence,  Z  =  5  or   —  3,    and    n=zQ  oy  10. 

3.  Given  a  =  275,   I  =  5,   and  n  =  46,  to  find  d  and  s. 

j^=-6, 
^''''    \s=       6440. 

4  Given  d=z6,  n  =  S,  and  s  =  156,  to  find  a  and  I. 

^"^^  I  "=3?: 

5.   Form  an  A.  P.  of  6  terms  whose  extremes  shall  be  7  and  37. 

Ans.  7,  13,  19,  25,  31,  37. 


Ans.   {"J 


328  SERIES. 

6.  Given  a  =  3,  n  =  CO,  and  s  =  3720,  to  jQnd  d  and  I 

=     2, 

121. 

7.  "What  is  the  sum  of  the  terms  of  an  A.  P.  formed  by  insert- 
ing 9  arithmetical  means  between  9  and  109  ?  Aiis,  649. 

8.  Find  the  sum  of  the  first  n  terms  of  the  progression  1,  2,  3, 
4,  5,  6,  ...  .  .  w  /I    ,      \ 

'     '  A71S.   S  =  -(l  4-  71). 

9.  Find  the  sum  of  the  first  n  terms  of  the  progression 
1,  3,  5,  7,  9,  ...  .  Ans.  s  =  w^. 

10.  Sum  to  30  terms  the  progression  116,  108,  100,  ...  . 

Ans.  5  =  0. 

11.  Sum  to  n  terms  the  progression  9,  11,  13,  15,  .  .  .  . 

Ans.  s  =n(8  -^  n). 

12.  Are  the  squares  of  a^  —  2x  —  1,  x^  +  1,  and  x^  -\-  2x  —  1 
in  A.  P.  ? 

13.  A  sets  out  from  a  place  and  travels  1  mile  the  first  day, 
2  the  second,  3  the  third,  and  so  on.  Five  days  later  B  sets  out 
from  the  same  place  and  travels  12  miles  a  day  in  the  same  direc- 
tion as  A.    How  long  will  A  travel  before  he  is  overtaken  by  B  ? 

A  US.  8  or  15  days. 

14.  A  sets  out  from  a  place  and  travels  1  mile  the  first  day, 

2  the  second,  3  the  third,  and  so  on.  B  sets  out  a  days  later  from 
the  same  place  and  travels  b  miles  a  day  in  the  same  direction  as  A. 
How  long  will  A  travel  before  he  is  overtaken  by  B  ? 

Show  that  B  will  never  overtake  A  if  a  >  ^^ — qT~^' 

15.  A  sets  out  from  a  place  and  travels  1  mile  the  first  day, 

3  the  second,  5  the  third,  and  so  on.  B  sets  out  three  days  later 
from  the  same  place  and  in  the  same  direction  as  A,  and  travels 
12  miles  the  first  day,  13  the  second,  14  the  third,  and  so  on. 
How  long  will  A  travel  before  lie  is  overtaken  by  B  ? 

Ans.  6  or  12  days. 


ARITHMETICAL    PKOGRESSIOIs\ 


329 


16.  The  distance  from  P  to  Q  is  165  miles.  A  sets  out  from 
P  toward  Q  and  travels  1  mile  the  first  day,  2  the  second,  3  the 
third,  and  so  on.  At  the  same  time  B  sets  out  from  Q  toward  P, 
and  travels  20  miles  the  first  day,  18  the  second,  16  the  third,  and 
so  on.    When  will  they  meet  ^ 

A?is.  At  the  end  of  10  or  33  days. 

Do  A  and  B  meet  twice  ? 

ARITHMETICAL    MEAK. 

508.  T7ie  Arithmetical  Mean  of  two  or  more  quan- 
tities is  the  quotient  obtained  by  dividing  their  sum  by  their 

number.    Thus,  the  arithmetical  mean  of  a  and  Z>  is    — - — ,  and 

2 

the  arithmetical  mean  of  1,  7,  11,  and  5  is  6. 

509.  To  find  the  arithmetical  mean  of  the  terms 
of  an  A.  P. 

Denoting  the  arithmetical  mean  by  M,  we  have,  by  definition, 

M  =  i. 
n 


But 


't     /  -TV 


M  = 


a  -f  I 


510.  To  find  a  and  I  when  M,  d,  and  n  are  given. 
s      {n-l)d 


a  = 


s      (n^l)d 
^-^+        2 


(506,  8). 


Substituting  M  for   -,  we  have 


a  =  M- 


l  =  M  + 


(n  —  l)d 

2 
{n  —  l)d 


330 


SERIES. 


511. 


PBOBZEMS. 


1.  Find  five  numbers  in  A.  P.  whose  sum  is  25,  and  whose  con- 
tinued product  is  945. 

Denote  the  arithmetical  mean  by  M  and  the  common  differ- 
ence by  X ;  then 


and 


M  =-^  =  0 
5 


the  first      term  =  5  —  2a;  ^ 
the  second  term  =z  5  —   x 
the  third    term  =  5 
the  fourth  term  =  5  +    x 
the  fifth      term  =  5  +  2a; 


(510); 


.-.     (5  -  2a;)  (5  -  a;)  5  (5  +  x)  (5  +  2a;)  =  3125  -  625a;3  +  20a;* 

=  945; 
1 
whence, 

The  required  numbers  are  therefore  1,  3,  5,  7,  9, 
or    5  —  a/109,     5  —  I  VT09,     5,     5  +  |  vT09,     5  +  vT09. 

2.  Find  four  numbers  in  A.  P.  whose  sum  is  32,  and  the  sum 
of  whose  squares  is  276.  Ans.  5,  7,  9,  11. 

3.  Find  three  numbers  in  A.  P.,  the  sum  of  whose  squares  is 
1232,  and  the  square  of  whose  arithmetical  mean  exceeds  the 
product  of  the  extremes  by  16.  Ans.  16,  20,  24. 

4.  Find  four  numbers  in  A.  P.  whose  sum  is  28,  and  whose 
continued  product  is  585.  Ans.  1,  5,  9,  13. 

5.  The  sum  of  the  squares  of  the  first  and  last  of  four  numbers 
in  A.  P.  is  200,  and  the  sum  of  the  squares  of  the  second  and  third 
is  136:  find  the  numbers.  Ans.  2,  6,  10,  14. 

6.  Find  the  first  term  and  the  common  difierence  in  an  A.  P. 
of  eighteen  terms,  in  which  the  sum  of  any  two  terms  equidistant 
from  the  extremes  is  31J,  and  the  product  of  the  extremes  is  85  J-. 


A         J  «  =3, 
^^^-    \d=li. 


GEOMETRICAL    PEOGRESSION.  331 

GEOMETRICAL    PROGRESSION. 

512.  A  Geotiietrical  Progression,  or  a  JPrO" 
gression  by  Quotient^  is  a  series  in  which  the  quotient 
obtained  by  dividing  the  second  term  by  the  first  is  equal  to  the 
quotient  obtained  by  dividing  any  other  term  by  the  preceding 
one.     Thus,  1,  3,  9,  27,  81  is  a  geometrical  progression. 

A  geometrical  progression  is  sometimes  called  a  Geometric 
cal  Series. 

For  brevity  we  shall  sometimes  use  G.  P.  for  the  phrase 
geometrical  progression. 

513.  TJie  Extremes  of  a  G.  P.  are  the  first  term  and  the 
last  term ;  the  other  terms  are  the  Means, 

514.  The  Hatio  of  a  G.  P.  is  the  quotient  obtained  by 
dividing  any  term  by  the  one  which  precedes  it.  Thus,  in  the 
progression  1,  3,  9,  27,  81  the  ratio  is  3. 

515.  An  Increasing  G,  JP»  is  one  in  which  the  ratio  is 
gi-eater  than  1.    Thus,  1,  2,  4,  8, 16  is  an  increasing  G.  P. 

516.  A  Decreasing  G,  JP,  is  one  in  which  the  ratio  is 
less  than  1.    Thus,  64,  16,  4,  1,  ^  is  a  decreasing  G.  P. 

51*7.  An  Infinite  Decreasing  G»  JP.  is  one  in  which 
the  ratio  is  less  than  1,  and  the  number  of  terms  is  infinite. 

518.  dotation.  In  treating  geometrical  progressions  we 
shall  use  the  following  notation : 

a  =  the  first  term  of  the  progression, 

I  =  the  last  or  n^  term, 

r  =  the  ratio, 

n  =  the  number  of  terms, 

8  =  the  sum  of  all  the  terms. 

Thus,  in  the  G.  P.        1,  3,  9,  27,  81, 

a  =  l,    Z  =  81,    r  =  3,    w  =  5,    s  =  121. 

519.  To  find  I  when  a,  r,  and  n  are  given. 

The  first  term  is  «,  the  second  term  is  ai^^  the  third  term  is  ar% 


832  SEBIES. 

the  fourth  term  is  ar^,  and  so  on;  hence  the  n^  term  is  ar»-i; 
that  is, 

I  =  ar""i  =  -  .  r*». 

7' 

Cor. — If  n  =  ao    and  r  <  1 :    then  l  =  -  x  0  =  0;  that  is, 

r 

The  last  term  of  an  infinite  decreasing  O.  P.  is  0. 

530.  To  find  s  when  a,  I,  and  r  are  given. 

5  =  rt  +  ar  +  a?-2  +  ar^  +  .  .  .  .   +  ar"~^    .     .     .     (1). 
Multiplying  (1)  by  r, 

r5  =  ar  +  ar^  4-  ar^  +  ffr*  4- .  .  .  .  -f  ar^    .    .    .     (2). 
Subtracting  (1)  from  (2), 

rs  —  s  =  ar^  —  a    ...     (3) ; 

whence,  5  = -r-    .    .     .     (4). 

r  —  1  ^ 

Substituting  /r  for  ar^  (519),  (4)  becomes 

Ir  —  a  ... 

r  —  1  ^  ' 

Cob. — K  7i  =  oo   and  r  <  1 ;    then  Z  =  0  (519,  Cor.),  and 

(5)  becomes  s  =  ———    .    .    .     (G) ;     that  is, 

TJie  sum  of  the  terms  of  an  infinite  decreasing  O.  P.  is  equal 
to  the  quotient  obtained  by  dividing  the  first  term  by  1  minus  the 
ratio. 

531.  In  a  G.  P.  the  product  of  any  two  terms  equidistant 
from  the  extremes  is  equal  to  the  product  of  the  extremes. 

Let  X  denote  a  term  which  has  m  terms  before  it,  and  y  a  term 
which  has  m  terms  after  it ;  then 


(519); 
whence,  xy  =  al. 


rx  =  ar"^     "I 


GEOMETRICAL    PROGRESSION.  333 

522.  To  insert  any  number  of  geometrical  means 
between  two  given  quantities. 

Let  a  and  h  be  the  given  quantities,  and  let  it  be  required  to 
insert  m  geometrical  means  between  them;  tbat  is,  let  it  be 
required  to  form  a  G.  P.  whose  extremes  are  a  and  b  and  the  num- 
ber of  whose  terms  is  m  +  2. 

Substituting  h  for  I  and  m  -\- 2  for  n  in  the  formula  of 
Art.  519,  we  liave 

whence,  r  =   y  -. 

By  multiplying  a  by  the  ratio  we  obtain  the  second  term ;  by 
multiplying  the  second  term  by  the  ratio  we  obtain  the  thu'd  term ; 
and  so  on. 

Example. — Insert  three  geometrical  means  between  7  and  112. 

hence  the  required  progression  is  7,  14,  28,  56, 112. 

523.  To  find  the  continued  product  of  the  terms  of 
a  G-.  P. 

Denoting  the  required  product  by  P,  we  have 

P  =  «  X  ar  X  «r2  X  ar^  X  .  .  .  .?    .    .    .     (1). 
Inverting  the  order  of  the  factors  in  the  second  member  of  (1), 
we  have 

P  =  ?  X  .  .  .  .  «r3  X  ar^  X  «r  X  a    .    .    •    (2). 
Multiplying  (1)  by  (2), 
P2  =  (al)  (al)  {al)  .  .  .  .  (al)  =  {aiy  ....  (3)  (521) ; 


whence,  P  =  V(«0"    •     •     •     (^)- 

524.  To   find   any  two  of  the   quantities  «,  I,  r, 
and  s  when  the  three  others  are  given. 

i  I  —  ar""-^   ) 
The  group  j     _  Ir-^a  > 


334 


SERIES. 


contains  the  five  quantities  a,  I,  r,  n,  and  s ;  hence  any  two  of 
them  may  be  found  when  the  three  others  are  given. 

Tlie  ten  cases  are  given  in  the  following  table  as  an  exercise 
for  the  student.  The  value  of  n  in  the  last  four  cases  cannot  be 
found  without  a  knowledge  of  the  properties  of  logarithms.  This 
part  of  the  work  must  therefore  be  deferred  until  Chapter  XXI 
shall  have  been  read. 


NO. 

eiviui 

Tonin> 

BEStTLTING  rORMUL^. 

1 

a,r,w 

l,s 

l-ar-^    «_«^'"-« 

i-ctr     ,o_   ^._^  . 

2 

I,  r,n 

a,s 

I             I  (r-  -  1) 

3 

n,r,  s 

a,  I 

s(r-l)    ,      (r_l).9r»»-i 
^  -  >«  _  1  '  ^  -       r"  -  1      • 

4 

a,  I,  n 

r,  s 

1               "           *» 

5 

a,  w,  s 

r,l 

ar»  —  rs  =  a  —  s,  I  (.9— Z)"-i=  n{s—a)''-K 

0 

I,  n,  s 

a,r 

a(s-  «)»-!  =  l(s  —  0"-S  (5-.Z)r»— 5r"-i 

7 

a,  r,  I 

s,  n 

Zr  —  «            log  Z  —  log  «  ,   ^ 
r  —  1 '                 log  r 

8 

a,  Z,  s 

r,  n 

5  — a                  log  ?  — log  « 

^^-5-  r  "-log(5-«)-log(s-Z)  ' 

9 

a,r,  s 

Z,  n 

«  +  5(r-l) 

log  [a  +  5(r-l)]-log«. 
log  r 

10 

I,  r,  s 

a,  n 

a  =  ?r  —  s  (r  —  1), 

logi-log[Zr-.(r-l)] 
^"-                    logr                    ^^- 

GEOMETRICAL    PKOGnESSIOlT.  335 


525,  pjtoiiLEJis. 

1.  The  first  term  of  a  G.  P.  is  3,  the  ratio  is  3,  and  the  number 
of  terms  is  12.    Find  the  last  term  and  the  sum  of  the  terms. 

We  have  given  a  =  3,  r  =  2,  n  =  12; 

?  =  3  X  2^^  =  6144, 
and  s  =  ^  ^^^~^  =  12285  (524, 1). 

2.  Given  s  =  1820,  w  =  6,  and   r  =  3,  to  find  a  and  I 

1820(3-1)       ^ 
^  =  -    36-1       =^> 

.      1820(3-^1)35      ^^_  __.    ^. 
and  I  = 36iri~  —  ^^^^  (^^*>  ^)- 

After  finding  the  value  of  «,  we  might  have  found  the  value 
of  I  from  the  formula  I  =  ar""^. 

3.  Find  the  sum  of  an  infinite  decreasing  G.  P.  of  which  the 
first  term  is  1,  and  the  ratio  -. 

s  =  ^-^  =  3  (520,  COE.). 

4.  Given  a  =  1,  /  —  512,  and  s  =  1023,  to  find  r. 

Ans.  r  =  2. 

5.  Insert  two  geometrical  means  between  24  and  192. 

Ans.  24,  48,  96,  192. 

3        9       27  1 

6.  Multiply    l+^  +  jg  +  ^+....to  infinity  by    -  — 

i  +  l55-6i+----*^^^^^^*y-  ^^''''i 

7.  Find  the  value  of  x  in  the  equation 

l-\-x-{'a^-\-0!^  +  9:^-{-a^-\-  ....to  infinity  =  2. 

Ans»  X  =  s- 


836  SERIES. 

8.  Find  the  ratio  of  an  infinite  decreasing  G.  P.  of  which  the 

5  1 

first  term  is  1,  and  the  sum  of  the  terms  -.  Ans,  r  =  -. 

4  5 

9.  Find  the  fii'st  term  of  an  infinite  decreasing  G.  P.  of  which 

the  ratio  is  — ,  and  the  sum  of  the  terms -— .      Ans.  a  =  l. 

m  m  —  1 

10.  Find  the  sum  of  the  first  n  terms  of  the  G.  P.  whose  m^ 

tennis  (— l)'»fl*^.                    .                  «*     r/      i\n  i»      -n 
^       ^  Ans,s  =  — -[(— l)»a^— 11. 

11.  Find  the  ratio  of  an  infinite  decreasing  G.  P.,  in  which 

each  term  is  ten  times  the  sum  of  all  the  terms  which  follow  it. 

.  1 

Ans.  r  =  —. 

12.  Find  the  sum  of  the  first  n  terms  of  a  G.  P.  whose  first 
term  is  a,  and  third  term  c.  ^  /  c^ 

a' 


Ans,  s  = 


fi- 


GEOMETRICAL    MEAN". 


526.  Hie  Geometrical  Mean  of  n  quantities  is  the 
n^  root  of  their  product.  Thus,  the  geometrical  mean  of  a  and  b 
is  Vab,  and  the  geometrical  mean  of  1,  3,  6,  and  72  is  6. 

527.  To  find  the  geometrical  mean  of  the  terms  of 

a  a.  p. 

Denoting  the  geometrical  mean  by  M,  and  the  product  of  the 
terms  by  P,  we  have,  by  definition, 

m  =  Vp. 


But  P  =  V{al)^  (523) ; 

M  =  Vol. 


GEOMETRICAL    PROGEESSIOX. 


337 


538.  To  find  a  and  I  when  M,  r,  and  n  are  given. 

I  al 


a  = 


ar 


n-l 


I 


^^n-1  _ 


alr*"-^ 


(519). 


Substituting  M^  for  al,  we  have 


a  = 


whence, 


a  = 


I 
M 


U  =  M  V^""^  J 


Cor.  1. — If  M  =  ^/~xy  and  r  =  ttHj,  all  the  terms  of  the 
progression  are  of  the  first  degi'ee  and  rational  if  n  is  even. 

The  sura  of  the  exponents  in  the  ratio  TT'^y  is  0 ;  hence  all 
the  tenns  are  of  the  same  degree.  The  ratio  x~^y  is  rational; 
hence  the  terms  are  either  all  rational  or  all  irrational.  It  is 
sufficient,  therefore,  to  show  that  the  first  term  is  of  the  first  degree 
and  rational. 


M 


n   2— n 


which  is  of  the  first  degree  and  rational  when  n  is  even. 

Cor.  2. — If  M  =  icz^  and  r  =  ar^y,  all  the  terms  of  the  pro- 
gression are  of  the  second  degree  and  rational  if  n  is  odd. 


a=z 


M 


x^y 


xy         ___ 


U|/ 


a^;/2 


Xi--nyn-l 


=  V  a:"+y~" 


n+l    ?— n  w+l        n— i 

=  x  ^  y  ^  =^X  ^  y      '^  , 


which  is  of  the  second  degree  and  rational  when  n  is  odd, 
22 


838  SERIES. 

539.  rjtoBZEMs. 

1.  Find  the  first  term  of  a  G.  P.  of  three  terms  whose  geo- 

y 

metrical  mean  is  xy  and  ratio  ^. 

Substituting  3  for  n,  xy  for  M,  and  -  for  r  in  the  formula 

M  ,  xy         xy        . 

a  = ,        we  have        a  =  — -==  =  -^=a*. 

2.  Write  a  G.  P.  of  three  terms  whose  geometrical  mean  is  xy 
and  ratio  -.  Ans.  os^,  xy,  y\ 

X 

3.  Write  a  G.  P.  of  four  terms  whose  geometrical  mean  is 

ni  Qfyi  nut 

\xy  and  ratio  -.  Ans.  — ,  x,  y,  ^. 

X  y  X 

4.  Write  a  G.  P.  of  five  terms  whose  geometrical  mean  is  xy 

and  ratio  -.  Ans,  — ,  x^,  xy,  y\  '—. 

X  y  ^    ^     X 

5.  Write  a  G.  P.  of  six  terms  whose  geometrical  mean  is  Vxy 

...    V  A       ^   ^  y^   y^ 

and  ratio  ^.  -  Ans,  -5,  — ,  x,  y,  ^,  ^. 

X  y^    y         ^    x'  x^ 

6.  The  sum  of  three  numbers  in  G.  P.  is  26,  and  the  sum  of 
their  squares  is  364.    What  are  the  numbers  ? 

Denote  the  geometrical  mean  by  xy  and  the  ratio  by  - ;  then 

by  the  conditions  of  the  problem, 

a^^xy  +y^=    2Q     .    ,    ,     (1), 

and  x^ -}- xy -\- y*  =z  364c    .    .    .     (2). 

Transposing  xy  in  (1)  and  squaring  the  result, 

x^ -{- 2x^y^ -\- y*  =  67Q  —  62xy -{- xiy^    .    .     .     (3) ; 

whence,       x*  -}-  x^y^  -\-  y^  =z  676  ^  52xy    .    .     .     (4). 


GEOMETRICAL    PROGRESSION-.  339 

Since  the  first  members  of  (2)  and  (4)  are  identical, 
364  =  676  —  520:^    .     .     .     (5) ; 

whence,  y  =-    .    .    •     (6). 

Substituting  this  value  of  y  in  (1)  and  solving  the  resulting 
equation,  we  find  a:^  _  ig  or  2. 

Squaring  (6)  and  substituting  for  x^  its  value,  we  find 

y^=2  or  18. 

From  (6)  xy  =  6. 

Hence  the  numbers  are  18,  6,  and  2. 

7.  The  sum  of  four  numbers  in  G.  P.  is  15,  and  the  sum  of 
their  squares  is  85.    What  are  the  numbers  ? 

Denote  the  geometrical  mean  by  V^cy  and  the  ratio  by  - ;  then 
by  the  conditions  of  the  problem, 

.15     .     .    .     (1), 

and  ^+a;2H.«/2  +  ^  =  85    .    .     .    (2). 

Assume  x  +  y  =z  z,  and  xy  =zp;    then 

x^  -\.  y'i  z=i  z^  —  2/7,     and    a?  -^  y^  :=^  z^  —  3zp. 
Substituting  2;  for  x  -^  y  in  (1)  and  z^  —  2p  for  x^  -^  y^  in 


y 

■\-x  -\-y  + 

t 

X 

+  a;2  +  «/2  + 

(2), 


|+.  +  f  =  15    .    .    .    (3), 


and  ^  +  ,»_2;,  +  g  =  85.  .     .     .     (4). 

Transposing  z  in  (3)  and  ^  —  %p  in  (4), 

^  +  -^  =  15-«    .    .    .    (5), 

and  J  +  g=:85-««  +  2^    .    .    .    (6). 


540  SERIES. 

Squaring  (5)  and  transposing  2xy  or  2/7, 

p+|^  =  (15-^)^-2i'    .    .    .     (7). 

Since  the  first  members  of  (6)  and  (7)  are  identical, 

(15  —  zy^%pz=^b  —  z^-\.2p    .    .    .    (8) ; 

whence,  2z^  —  30^;  —  4p  =  —  140    .    .    .    (9). 

Clearing  (5)  of  fractions, 

7?  ■\-  if  =  {lb  —  z)  xy  =  15/7  —pz    .    .    .     (10). 

Substituting  z^  —  3zp  for  ofi  +  y%   (10)  becomes 

z^  —  3zp  =  15p—pz    .    .     .     (11); 

z^ 

whence,  p  =  — —    .    .    .    (12). 

-^10  +  2z  ^     ' 

Substituting  this  value  of/?  in  (9),  clearing  of  fractions,  trans- 
posing and  reducing,  we  obtain 

15^3  +  85;?  =  1050    .     .    .     (13) ; 

35 

whence,  2;  =  6   or —, 

o 

Substituting  6  for  z  in  (12),  we  find 

^  =  8. 

We  then  have  the  equations 

ic  +  y  =  6    .    .    .     (14), 

and  xy=%    .    .    .     (15) ; 

whence,  a;  =  4  or  2,    and    y  =  2  or  4. 

The  raquired  numbers  are  therefore  1,  2,  4,  8. 
The  second  value  of  z  leads  to  imaginary  results. 
In  the  solution  of  such  problems  as  this  and  the  preceding  one, 
the  terms  of  the  progression  may  be  represented  by 

X,  xy,  xy^,  xif,  a;^,  .  .  .  . 

but  the  notation  we  have  used  is  generally  preferable. 


THE    DIFFEKEKTIAL    METHOD.  341 

8.  The  sum  of  three  numbers  in  G.  P.  is  210,  and  the  last  ex- 
ceeds the  first  by  90.    What  are  the  numbers  ? 

Ans.  30,  GO,  120. 

9.  The  continued  product  of  three  numbers  in  G.  P.  is  216, 
and  the  sum  of  the  squares  of  the  extremes  is  328.  What  are 
the  numbers?  A^is.  2,  6,  18. 

10.  The  continued  product  of  three  numbers  in  G.  P.  is  G4, 
and  the  sum  of  their  cubes  is  584.    What  are  the  numbers  ? 

Ans.  2,  4,  8. 

11.  The  sum  of  120  dollars  was  divided  among  four  persons  in 
such  a  manner  that  the  shares  were  in  A.  P.  If  the  second  and 
third  had  each  received  12  dollars  less,  and  the  fourth  24  dollars 
more,  the  shares  would  have  been  in  G.  P.     Find  the  shares. 

Ans.  $3,  $21,  $39,  857. 

12.  The  sum  of  six  numbers  in  G.  P.  is  189,  and  the  sum  of 
the  third  and  fourth  is  36.     What  are  the  numbers  ? 

Ans,  3,  6,  12,  24,  48,  96. 

TREATMENT  OF  SERIES  BY  THE  DIFFERENTIAL  METHOD. 

530.  The  First  Order  of  I>ifferences  of  a  series 
is  the  series  obtained  by  subtracting  each  term  of  the  given  series 
from  the  following  term ;   the  Second  Order  of  l>fffer~ 

ences  is  the  series  obtained  by  subtracting  each  term  of  the 
first  order  of  difierences  from  the  following  term ;  tlie  Third 
Order  of  Differences  is  obtained  from  the  second  in  the 
same  way  as  the  second  is  from  the  first ;  and  so  on.    Thus, 

If  the  given  series  be  1,  4,  9, 16,  25,  .... 

The  1st  order  of  differences  is      3,  5,    7,    9,  .  .  .  . 
The  2d  order  of  differences  is  2,    2,    2,  .  .  .  . 

The  3d  order  of  differences  is  0,    0,  ...  . 

531.  The  Differential  31ethod  is  the  process  of  find- 
ing any  term  of  a  series,  or  the  sum  of  any  number  of  its  terms, 
by  means  of  the  successive  orders  of  differences. 


342  SEKIES. 

532.  To  find  the  n^^  term  of  a  series. 

Let  a,  h,  Cf  d,  6)  ,  ,  ,  ,  be  the  proposed  series. 
The  1st  order  of  differences  is  J— a,  c—h,  d—c,  e^d, .... 

The  2d  order  of  differences  is  c — 2b  +  a,  d—2c  +  5,  e— 2c?4-  c, . . . . 
The  3d  order  of  differencesis  d—dc-^Sb—ay  e—dd-\-3c—bf . . . . 
The  4th  order  of  differences  is  e — 4e? + 6c — 45  +  «, . . . . 


Denote  the  first  term  of  the  first  order  of  differences  by  di,  the 
first  term  of  the  second  order  of  differences  by  d^^  the  first  teim 
of  the  third  order  of  differences  by  d^,  and  so  on ;  then 


c?i  =:  J  —  a 

di-=c  —2b  -\-  a 

d3  =  d  —  Sc  +  db  —  a 

di  =  e  —  M  -\-  6c  —  4:b  -{-  a 


I. 


•  .   (1); 


whence, 


'  b  z=  a  +  di 
c  =z  a  -}-  2di  -i-  di 
d  =  a  -\-3di-\-  dd^  +  ds 
e  =  a  -\-  4:di -\'  6di -^  ^dz  +  d^, 


(2). 


The  coefficients  in  the  value  of  c,  the  tliird  term  of  the  pro- 
posed series,  are  1,  2,  1,  which  are  the  coefficients  of  the  expan- 
sion of  {x  +  of ;  the  coefficients  in  the  value  of  d,  the  foiirth 
term,  are  1,  3,  3,  1,  which  are  the  coefficients  of  the  expansion  of 
{x  +  of ;  the  coefficients  in  the  value  of  e,  the  fifth  term,  are 
1,  4,  6, 4, 1,  which  are  the  coefficients  of  the  expansion  of  {x-\-aY\ 
and  so  on.  Hence  the  coefficients  in  the  value  of  the  n^^  term  are 
the  coefficients  of  the  expansion  of  {x  +  aY~\  Therefore,  denot- 
ing the  n*^  term  of  the  series  by  T„, 

T„= 


flf  +  (w— 1)^1  + 


in- 


l)(n-2)^^  ^  (n-l)(n-2)(^ 


3) 


la 


^ 


^3+....  (A). 


THE    DIFFERENTIAL    METHOD.  343 


EXAMTIjES. 

1.  Find  the  12th  term  of  the  series  1,  4,  9,  16,  25,  ...  . 

In  this  example  a  =  1,  J,  =  3,  tZg  =  2,  d^  =  0,  and  w  =  12. 
Substituting  these  values  in  (A),  we  obtain 

T.  =  l  + 11x3 +  11^115^  =144. 

2.  Find  the  9th  term  of  the  series  1,  4,  8,  13,  19,  ...  . 

Alls,  53. 

3.  Find  the  15th  term  of  the  series  1,  4,  10,  20,  35,  .... 

Ans,  680. 

4.  Find  the  8th  term  of  the  series  1,  6,  21,  56,  126,  251,   . 

Ans,  771. 

5.  Find  the  20th  term  of  the  series  1,  8,  27,  64,  125, 

Ans.  8000. 

6.  Find  the  n^  term  of  the  series  1,  3,  6,  10,  15,  21,  .... 

Ans,  ^J^, 
1%  n{n-\-  1)  divisible  by  2  ?    Why  ? 

7.  Find  the  n^  term  of  the  series  1,  4, 10,  20,  35,  ...  . 

Ans.  "(«  +  l)(»  +  ^). 
6 
Is  w  (n  +  1)  {n  +  2)  divisible  by  6  ?    Why  ? 

8.  Find  the  n^  term  of  the  series  1,  5,  15,  35,  70,  126,  .  .  .  . 

Ar,.   n{n  +  l){n  +  %)(n^^) 

Is  w  (71  +  1)  {n  +  2)  (;i  4-  3)  divisible  by  24  ?    Why  ? 

533.  To  find  the  siim  of  n  terms  of  a  series. 

Let  a,  h,  c,  d,  e,    .    .    .     (1) 

DC  the  proposed  series,  and  denote  the  sum  of  n  terms  of  it  by  S, 
Let  us  assume  the  series 

0,  a,  a  -\-  h,  a  -{•  l  +  c,  a  +  1)  +  c  +  dy    .    .    .    (2). 


344  SERIES. 

Now  it  is  evident  that  the  sum  of  n  terms  of  (1)  is  equal  to 
the  {n  4-  1)^  term  of  (2). 

Denoting  the  {n  -[-  1)^  term  of  (2)  by  T^+i,  the  first  term  of 
the  first  order  of  differences  of  (2)  by  D„  the  first  tenn  of  the  sec- 
ond order  of  differences  by  Dg,  and  so  on,  we  have  by  (A), 

T.,.=0+«D,+fcllD,+^i^%.+ (3). 

But  T,+i=S„,  D,  =  «,  J)i=  b  —  a=id^,  D3=  c—  25  +  «  =  f?2, 
and  so  on.    Hence,  by  substitution,  (3)  becomes 

S,=yzfl+   ^  .,^    Vi4-— ^ -d,+ (B). 

EXAMPLES, 

1.  Find  the  sum  of  10  terms  of  the  series  1,  4,  9, 16,  25, .  .  .  . 
In  this  example  cr  =  1,  ^,  =  3,  c?,  =  2,  dz  =  0,  and  w  =  10. 
Substituting  these  values  in  (B),  we  obtain 

10  X  9  X  3       10  X  9  X  8x2 

2  "^  6 

2.  Find  the  sum  of  20  terms^of  the  series  1,  3,  6, 10, 15,  21, ... . 

Ans.  1540. 

3.  Find  the  sum  of  12  terms  of  the  series  1, 5, 14, 30,  55, 91, 

Ans.  23G6. 

4.  Find  the  sum  of  10  terms  of  the  series  1, 4, 13, 37, 85, 1 GG, ... . 

Ans.  2755. 

6.  Find  the  sum  of  n  terms  of  the  series  1,  3,  6,  10,  15,  21, ... . 

Ans.  M!L±iHfi±2)_ 
6 

INTERPOLATION. 

534.  Interpolation  is  the  process  of  inserting  between 
two  consecutive  terms  of  a  given  series  a  term  or  terms  which 
shall  conform  to  the  law  of  that  series. 

535.  The  Formula  for  Interpolation  is  that  given 
for  finding  the  n^^  term  of  a  series  by  the  differential  method. 


S,„  =  10  +  ^"   .^^  ""  +        "     r  =  385. 


THE    DIFFERENTIAL    METHOD. 
EXAMPLES. 


345 


^  V2I  =  2.758924  ' 
V22  =  2.802039 


Given   \  y^ 

V24 


2.843867 
2.884499 
L  V25  =  2.924018 


to  find  the  cube  root  of  any  in- 
termediate number  by  the  differ- 
ential method. 


1.  Find  the  cube  root  of  21.75. 

The  operation  of  finding  the  orders  of  differences  may  be  con- 
veniently arranged  as  follows : 


NO. 

CUBK  ROOTS. 

d. 

d. 

d. 

d. 

21 

2.758924 

22 

2.802039 

+  .043115 

23 

2.8438G7 

+  .041828 

-.001287 

24 

2.884499 

+  .040C32 

-.001196 

+  .000091 

25 

2.924018 

+  .039519 

—.001113 

+  .000083 

—.000008 

The  distance  between  any  two  consecutive  terms  of  the  given 
series  is  1 ;  hence  the  value  of  n  which  corresponds  to  the  required 
term  is  If;  that  is,  the  required  term  is  J  of  the  way  from  the 
first  to  the  second  term.  Substituting  in  (A)  1|  for  n,  2.758924 
for  a,  .043115  for  d^,  —  .001287  for  ^g,  .000091  for  d^,  —  .000008 
for  d^,  and  reducing,  we  find  Ti|  =  V2r75  =  2.791385. 


2.  Find  the  cube  root  of  21.325. 

3.  Find  the  cube  root  of  21.875. 

4.  Find  the  cube  root  of  21.4568. 
6.  Find  the  cube  root  of  22.25. 

6.  Find  the  cube  root  of  22.684:. 

7.  Find  the  cube  root  of  22}. 


Arts.  2.773083. 
Ans.  2.796722. 
Ans.  2.778785. 
Ans.  2.812613. 
Ans.  2.830783. 
Ans.  2.833525. 


346  SERIES. 

DEVELOPMENT  OF  EXPRESSIONS  INTO  SERIES. 

536.  To  Develop  or  Expand  aa  expression  is  to  con- 
vert it  into  a  series. 

537.  To  develop  a  fraction  into  a  series  by  division. 


EXAMPLES, 


1.  Convert 


3rii 

1—X 

mto  an  inuniie  Beries. 

1 

1—X 

1  —  x 

l  +  a;  +  ic2  +  a^  +  etc. 

X 

x  —  a? 

a? 

a?-x^ 

a^ 

1  +x  -\-x^  r^x^  +  x^  -{■  etc.  to  infinity 

(1) 


If  a;  =  |,   (1)  becomes    2  =  l  +  |  +  ^  +  i  +  etc.  ...  (2) 

If  x  =  l,   (1)  becomes  oo  =  l  +  l4-l-f-l4-  etc.  ...  (3) 
If  x  =  2j   (1)  becomes  — 1  =  1  +  2  +  4  +  8  +  etc.  ...  (4) 
How  is  this  result  to  be  explained  ? 
Convert  each  of  the  following  fractions  into  an  infinite  series 

^         a  ,       ^       X      x^       x^       X* 

2.  — --.  Ans.  1 +  -2 -A  +  ~i—   ••• 

ri  -i-  ^  a      a^       a"^       a^ 

.        ^        X        x^        x^        X* 

3.     .  Ans,  1  +  -  +  -o  +  -3  +  -.  +    ... 

4.  ^— ^-^.  A71S.  l-\-2x  +  2x^-\-2a^-\-2x^+    ,  .  . 

■      1  X^    ,     X^         X^         X? 

Ans, «  +  — , 7  +  -9  —  •  •  • 

a       a^       a''       a'       a^ 

6. -.     Ans,  l  +  a—a^—a^^-a^-\-a^—a^—a>^+  ..  . 

1  —  a  +  a^ 


a  +  X 

a 

a  —  X 

l-{-x 

1  —  x 

a  -\-  X 

a^  +  x?' 

1 

DEVELOPMENT    OF    EXPRESSIONS    INTO    SERIES.  347 

538.    To    develop    an    expression   of  the   form   of 


Vm  ±,n  by  extracting  the  indicated  root. 


EXAMPLES. 


1.  Convert   a/1  +  ^  into  an  infinite  series. 

l  +  ^l  +  3-^  +  ^-j28  +  etc. 


^  +  5 


X 


2  4-a; 


8/         4 


a;2       ic3       ^4 
~T~"8  "^64 


2+^~l+fi) 


"8  ~6i 

"8  "*"  16  ~  64  "^  256 


""64  "^  64  ""  256 


X       ^       ^        hxS 


^^^^+^=^  +  2^8+16-128+  •'•• 

Convert  each  of  the  foUowing  expressions  into  an  infinite 
series : 


2.  \/a  —  X, 

i(         x_  x^  3a:3 3-5a^  \ 

^'^^-  "^  r~'2a-2-4a2  2-4-6a3      2-4-6-8a*                /* 

3.  V«M^. 

Z»2          ^4  3^_           3-5^8 


4.     Vl  -  X. 

X       x^  _  a^  _  5x*  _  Ix^ 
Ans.     1------      ^^g       ^^g 


348 


SERIES. 


539.  To  develop  an  expression  by  means  of  unde- 
termined coefficients. 


1  4-  2a: 

1.  Convert jr-  into  an  infinite  series. 

1  —  3a; 

1  -h2x 


Assume 


=  A  +  Ba;  +  Cie2  +  DaJ*  +  Ea:*  +  .  .  .  (1). 


1  — 3a; 

Multiplying  (1)  by  1  —  3a;,  we  obtain 


1  +  2a;  =  A  4-    B\x  +    C 
—  3a!    -3B 


a--2+   D 
-3C 


-3D 


a;*  +  .  .  .  (2). 


Equating  the  coefficients  of  like  powers  of  x  in  the  two  mem- 
bers of  (2), 

A  =  l  1 

B  -  3A  =  2 
C  -  3B  =  0 
D  —  3C  =  0 
E  —  3D  =  0 


whence,    A  =  1,    B  =  5,    C  =  15,    D  =  45,    E  =  135, 
Substituting  these  values  in  (1), 
1  +  2a; 


1  — 3a; 
2.  Convert 

Assume 


=  1  +  5a;  +  15a;2  ^  4^^^  ^  135^:1  +  .  . 


into  an  infinite  series. 


3x-j^ 

1 


=  A  +  Ba;  4-  Ca;2  +  I)a:3  ^  Ea;4  +  .  .  .  (ij. 


3x  —  a;2 
Multiplying  (1)  by  3a;  —  x^,  we  obtain 


1  =  3Aa;  +  3Bb2  +  3c 

-  Al     -   B 


a;3  +  3D 
-    C 


^  +  .  .  .  (2); 


whence,  1  =  0,  which  is  absurd ;  hence  the  second  member  of 
(1)  is  not  of  the  proper  form. 


Assume 


DEVELOPMENT    dF    EXPKESSIONS    INTO    SERIES. 

1 


349 


'dx  —  x^ 
Multiplying  (3)  by  Zx  —  x\ 

1  =  3A  +  3B|a;  +  3C 

-   aI    -    B 


=  Aic-i  +  Ba:0+Ca;+Da;8+Ea;3^.  .  .  (3). 


a;2  +  3D 
-    C 


7^-\-.   .   .   (4). 


Equating  the  coefficients  of  like  powers  of  x  in  the  two  mem- 
bers of  (4), 

3A  =  1  ' 
3B  —  A  =  0 
3C  —  B  z=  0 
3D  —  C  =  0 


whence,    A 


B 


Substituting  these  values  in  (3), 

■"  3a;  "^  9  "^  27  "^  81  "^'  •  •  • 

The  proper  form  of  the  second  member  of  the  assumed  identity 
may  be  detemained  in  each  case  by  observing  what  the  given 
expression  becomes  when  the  variable  is  supposed  to  be  zero.  If 
the  given  expression  becomes  a  finite  quantity,  the  first  term  of 
the  series  will  not  contain  the  variable ;  if  it  becomes  zero,  the  first 
term  of  the  series  will  contain  the  variable ;  and  if  it  becomes 
infinity  the  first  term  of  the  series  will  be  of  the  form  Ax"^. 

Convert  each  of  the  following  expressions  into  an  infinite  scries : 

1  o^ 

3. 5^.  ^ws.  l+a;+3a;2+9a;3  +  37a^+8l2:54..... 

1  —  6x 

1  4-  2r 

4.  = — ..     Ans.l  +  Zx+4.x^-\-W-\-llx^-\-l^x^-\-..>. 


5. 


l-^x  —  x^ 

\-^x 

l_3a;_2a;2 


Ans.  l  +  2a;  +  8a;2-f  28rr3  +  100a;^  +  356a;5^. 


350  SERIES. 
{L      4X) 

2  A  .  ^  .  ^''^  .  ^^^     3^^ 

^-    3^32^^-  ^''^•32;'^9  +  27'^"8r  +  2i3+*--- 


8. 


1 


1  +  22^^4- 3a;* 

Ans,  l—23^+x^+4:3f^—lla^-{-10x^-\-ldx^— 

RECURRING    SERIES. 

540.  A  Recurring  Series  is  one  which  may  be  pro- 
duced by  expanding  some  rational  fraction.     Thus, 

is  a  recurring  series,  because  it  is  the  expansion  of  the  fraction 

:j (537,  1).     In  this  series  all  the  terms  after  the  first  two 

recur  according  to  a  definite  law. 

541.  The  Generating  Fraction  of  a  recurring  series 
is  the  fraction  which  can  be  converted  into  the  given  series.  Thus, 
the  generating  fiuction  of  the  series    l-\-x-\-3x^-\-%a?-^ 27ic* 4- . . . 

.    1  — 2a; 


IS 


l-3a;' 


543.  In  the  series  given  in  Art.  540,  each  term  after  the 
first  may  be  obtained  by  multiplying  the  preceding  term  by  x ; 
and  in  the  series  1  +  42:  +  lla^^  +  34a:3  j^  loia:*  +  .  .  .  .  the 
sum  of  the  products  obtained  by  multiplying  the  first  of  any  two 
consecutive  terms  by  3x^  and  the  second  by  2a;  is  equal  to  the  next 
succeeding  term.  The  expression  by  means  of  which  any  term  of 
a  series  may  be  found  when  the  preceding  terms  are  known  is 
called  the  Scale  of  Relation,  Thus,  the  scale  of  relation  of 
the  series  l-^x-{-x^-\-3?-i^x^-\-.,,,  is  a;,  and  the  scale  of 
relation  of  the  series    1  +  4a;  + 1  la;^  +  34a:^  + .  .  .  .     is    3x^-\-2x. 

543.  A  recurring  series  is  said  to  be  of  the  n^  order  when  the 
number  of  terms  in  its  scale  of  relation  is  n.  Thus,  the  series 
1  +  4a;  +  lla;3  _|_  34^  +  .  .  .  .    is  of  the  second  order. 


RECUEEING    SEEIES.  351 

544.  To   find  the  scale   of  relation  in   a  recurring 
series. 

Let  a-{-b-{-c-{-d-\-e-\-,...   be  the  proposed  series. 

1st.  Suppose  that  the  series  is  of  the  first  order. 

Let  m  denote  the  scale  of  relation ;  then  b  =  ma ;    whence, 

b 

m  =  -. 

a 

2d.  Suppose  the  series  to  be  of  the  second  order. 
Let  m  +  71  denote  the  scale  of  relation ;  then 


i^=:::::i(^*^-^3); 


mb  -f-  7ic 

,  c^  —  bd  ,  ad  —be 

whence,  m  = j-„,     and     w  = rr-. 

ac  —  IF  ac  —  b^ 

3d.  Suppose  the  series  to  be  of  the  third  order. 
Let  m  -\-n  -{•  r  denote  the  scale  of  relation ;  then 

i  d  z=  ma  -{-  nb  -\-  re  \ 
<  e  ■=.  mb  -\-  nc  -{-  rd  > , 
\  f  =zmc  -\-  nd  -\-  re  ) 

From  this  group  of  equations  m,  n,  and  r  may  be  found. 
If  the  series  is  of  the  fourth  order,  fifth  order,  sixth  order,  &c., 
the  scale  of  relation  may  be  found  in  a  similar  manner. 

CoE.  1. — If  the  proposed  series  is  of  the  n^  order,  2n  con- 
secutive terms  must  be  given  to  enable  us  to  find  the  scale  of 
relation. 

CoE.  2. — If  we  assume  any  proposed  series  to  be  of  a  higher 
order  than  it  really  is,  one  or  more  of  the  terms  of  the  scale  of 
relation  will  be  found  to  be  equal  to  zero. 

If  we  assume  any  proposed  series  to  be  of  a  lower  order  than  it 
really  is,  or  if  we  attempt  to  find  the  scale  of  relation  of  a  series 
which  is  not  recurring,  the  error  will  appear  if  we  attempt  to  apply 
the  scale. 


SERIES. 


EXAMPLES. 

Find  the  scale  of  relation  in  each  of  the  following  series: 
1.  1  +  4a;  +  IQx"'  +  223^ -\- .  .  .  . 

Assume  the  scale  of  relation  to  he  m  +  n;    then 
J  10a;2  =  fn-\-  ^nx         )  ^ 

whence,  m  =  —  2a^,  and  n  =  32?.    Therefore  the  scale  of  rela- 
tion is   —  2j?5  +  3a;. 


2.  l4-6a;  +  12arJ  +  48a;3  +  120a:*+ 

3.  l4-2a;  +  3a;H4a:3_f_5a4^  .  .  . 

4.  l  +  2a;+8ir3+28j^+100a;*-h  . 


^ws.  6a;2_|_^, 

Ans,  —x^-\-2x, 

Ans.  2xi-^dx. 

Ans.  2x'''  +  2x. 


54:5.  To  find  the  generating  fraction  of  a  recurring 
series. 

Let  a-\-b-{-c-^d  +  e  +  ....   be  the  proposed  series. 
1st.  Suppose  the  series  to  be  of  the  first  order. 
Let  m  denote  the  scale  of  relation ;  then 

b  =  ma 
c  =  mb 
d=  mc 
e  =  md 


whence,  bi-c  +  d+e-{- .  .  .  .  =ni(a  +  b  +  c  +  d+e-^ ,  .  .  .). 

Hence,  denoting  the  generating  fraction  or  the  sum  of  the 
series  by  Fj,  we  have 


whence, 


^'-r:^ 


m 


(0,). 


EECURBINQ    SERIES. 


353 


2d.  Suppose  the  series  to  be  of  the  second  order. 
Let  m  -{-  n  denote  the  scale  of  relation ;  then 

c  =  ma  -f  nb 
d=:mb  -i-  no 
e  =  mc  +  nd 
f  =  md  -f  ne 


whence, 

Hence,  denoting  the  generating  fraction  by  Fg,  we  have 
Fg  -{a  +  l)  =  wFg  +  n(F^  -  a) ; 
^        a  +  1)  —  an 


whence. 


1  —  7/^ 


(0»). 


3d.  Suppose  the  series  to  be  of  the  third  order. 
Let  m  •{■  n  -{•  r  denote  the  scale  of  relation ;  then 

d  =  ma  -^  nh  -\-  re 
e  =imh  ■\-  nc  -\-  rd 
f  ^rnc  4-  nd  +  re 
g  =  md  ■\-ne  -^r  rf 


whence,  d+e+f-\-g-\-. . .  .=m  {a-\-h-\  c-\-d+e+f+g-\-. . . .) 

Hence,  denoting  the  generating  fraction  by  Fg,  we  have 
F3  --  {a  +  h  +  c)  =  m¥^  H   -^  (F3  -a)  +  r[F,-{a  +  b)] ; 
whence, 

F, 


a  -{-  b  -{■  c  —  an  —  (a  +  b)  r 
1  —  m  —  n  —  r 


■    (0,). 


If  the  series  is  of  the  fourth  order,  fifth  order,  sixth  order,  etc., 
the  generating  fraction  may  be  found  in  a  similar  manner. 

ScH. — The  formulae  fOj),  (O^),  (O3),  etc.,  have  been  obtained 
on  the  hypothesis  that  the  given  series  is  infinite  and  converging. 


354  SERIES. 

EXAMPIiES, 

Find  the  generating  fraction  of  each  of  the  following  series : 

1.  1  +  4a; +10a?J+ 322:34- 46a:* -f  .... 

In  this  series  the  scale  of  relation  is  —^x^-\-  3x  (544,  1) ; 
hence  (Og)  is  appUcable.  Substituting  —  23^  for  m,  3x  for  w, 
1  for  a,  and  4:X  for  b,  we  have 

-,        l+4a;—  3a;  l-fic 


^  ^  1  +  23^  —  3x~  1  -]-  23^  —  3x 

1  +  2a; 


2.  l  +  3a;-f4a^5+7a;3^11a4^__  j;^^. 

3.  l  +  6a;+12a;2^48a:3^120a:4-}-....       A7is, 

4.  l  +  2a:— 5ar^  +  26a;3— 119a:*+....       Ans. 


l  —  a;— a:2" 

1  +  5a; 
l_a;__6a;2" 

1  +  6a; 


1  +  4a;  —  Sxi' 
6.    l  +  4a;+3a;2_2a;3+4a4^X7a;5_j.3a4_.._ 

H-3a;  +  arJ 


Ans. 


l—x-\-2xi—3a^' 
1-^.x 


6.    lH-3a;4-5a;8+7a;8^9a4^  ,..,  Ans,,  .^. 

(1  —  a;) 

REVERSION  OF  SERIES. 

546.  To  Hevert  a  Series  containing  an  unknown  quan- 
tity is  to  express  the  value  of  that  unknown  quantity  in  terms  of 
the  sum  of  the  given  series.  Thus,  to  revert  the  series  in  the  sec- 
ond member  of  the  equation 

y  z=  ax  ■{■  hx^  -\-  CT^  •\-  dx^  +  ea;^  +  .... 

is  to  find  the  value  of  x  in  terms  of  y. 

EXAMPLES. 

1.  Revert  the  series  in  the  equation  y=zx-[-x^-\-x^-\-a^-\-  .... 

X 

This  is  a  recurring  series  whose  generating  fraction  is ; 

X.  •""  X 


REVEKSION    OF    SEEIES. 


355 


henco  y  = 


1-x' 


whence,  x  = 


y     ^ 


y  —  y^+y^—y^+ 


Q?  iC^  iC^ 

2.  Revert  the  series  in  the  equation  y=ix——-\r  t  —  -k-\- 


2x 


This  is  a  recurring  series  whose  generating  fraction  is  — - — 

/v   -p   X 


hence  y  = 


2x 


._  ^y  _, 


y^  ,  y^ 


2  H-  .;'  ^^^'^'''  ^=21:^=2/  + Y  +  t  +  t  + 


A  recurring  series  cannot  be  reverted  by  this  method  when 
the  equation,  formed  by  placing  y,  the  sum  of  the  given  series, 
equal  to  the  generating  fraction,  cannot  be  solved.  The  method 
used  in  the  following  example  is  applicable  to  any  series. 

3.  Revert  the  series  in  the  equation 

yz=ax-^la^  +  ca^-\-dx^-^  .... 

Assume    x=Ay-[-By^-\-Cy^  +  T)y^-{-  .... 

in  which  the  coefi&cients  A,  B,  C,  D,  .  . .  , 

Substituting  for  y  its  value,  (2)  becomes 


•    •     •     (!)• 
.    .    .    (2), 
are  undetermined. 


X  =  aAjx  +  bA 
+  a2B 


^2  + 

cA\ 

+  2a^B 

+ 

a^C\ 

+ 

dA 

+ 

hm 

-f 

2acB 

+  Sa^C 

+ 

a^D 

x^-^ 


(3). 


r«A  =  i 

bA  -\-  a^B  =  0 

cA  +  2abB  -\-  a^C  =  0 

dA  +  ^B  +  2acB  +  da^C  -f  a^D  =  0 


whence. 


a  0^  a? 

Substituting  these  values  in  (2), 
1         b    ^      2W'-ac 

^  =  -^y--^y^-^ — :;5— ^ 


aH—6abc-\-bl^ 


a^d—5abc-\-5b^ 


y+. 


(4). 


366  SERIES. 

4.  Eevert  the  series  in  the  equation 

i/  =  x-\-2x^  +  4^-\-Sx^-\-  .  .  .  . 
In  this  series 

«  =  1,    5  =  2,    c  =  4,  J  =  8,  .  .  .  . 

Substituting  these  values  in  (4)  of  the  preceding  example,  we 
have 

6.  Eevert  the  series  in  the  equation 

1_           4xi      63^      Sx^ 
-_Zx--^+   ^   -   ^   + 

and  find  the  value  of  x. 

Substituting  j  for  y,  2  for  «,  —  q  ^or  b,  -  for  c,  —  -  for  «/, . . .. 
4  o  o  7 

in  (4)  of  example  3,  we  have 

1\* 


\  &  '■'&  »©' 


+  .000013  +  .  .  .  .    =  .135993  +. 

6.  Eevert  the  series  in  the  equation 

y  =  x-\-dx^'^5a^  +»7a.-4  ^  Q3^  ^  ,  ,  ,  , 

Ans.  x  =  ij^dy^  +  13/  _  e7y^  +  381?/»  —  .  .  . . 

7.  Eevert  the  series  in  the  equation 

a^       x^       x^       a^ 

y2  jiS  jA  ^ 

8.  Eevert  the  series  in  the  equation 

y  =  X-\-7?  -^-X?  -^X^  ^-X^  -{■    .... 

Ans.  x=z  y  —  y^  -\-  2y^  —  by'  +  14/  —  .... 

9.  Find  the  value  of  x  in  the  equation 

I  =  5a;  —  20a;2  +  SOa:^  _  3202?^  +  I280a:5  _  .  .  .  . 

Ans.  2:  =  .117047+. 


BIKOMIAL    FORMULA.  357 

THE  BINOMIAL  FORMULA  FOR  ANT  EXPONENT. 

547,  It  has  been  shown  that  when  nis  q>  positive  integer 

71/  ( 11  —  1  ^ 

{x  +  «)«  =  x^  +  nax''-'^  +  — ^- — '  a2^«-2  +  .  .  .  . 

"We  now  proceed  to  show  that  this  formula  is  true,  whether  n 
\%  positive  or  negative,  entire  or  fractional, 

548.  Lem. — Hie  value  of —,  when  y  =  x,  is  nx'^~\ 

X  —  y 

whether  n  {^positive  or  negative,  entire  or  fractional. 

1.  When  w  is  a  positive  integer. 

The  proposition  has  been  shown   to  be  true  for  this  case 
(461,  Cor.  2). 

2.  When  nis  a.  positive  fraction. 

Let  n  = -,  in  which  j!?  and  q  are  supposed  to  be  positive  in- 

(    2         P) 

tegers.    We  are  to  show  that  1 —  \       z=:^x9     . 

^  i  ^  —  y   }y=x     q 

(  -Y    (  -Y 


p 

XQ 


p     {  ^\p     (  iV  1       1 

—  yq_\XQ/  —Kyi)  a:?  —  ;/g 

""^        \xl)  —  Ky'n)        \x'q)  —  \yl) 


1         1 

x'd  —  yi 


But,  by  the  first  case,     i  \x<i)-\y^)    I        ^^ (4)^; 

(  xl—  y"^  )   y=x 

(  XQ—yi  )  y=x 

(P  P\  (     1\7^1 


XI 


358  SERIES. 


3.  When  nia  a  negative  integer. 


Suppose  w  to  be  a  positive  integer  and  that  n  =  —  m.    We 
are  to  show  that    \ —  y       —  —  mar^-\ 


\       ^—y       \  y=x 

X  —  y  ^      \  x  —  y  r 

But,  by  the  first  case,    \  — ^^^  i       =  maf^-\ 

^  i     ^-y     )y=x 

( ^^^ — 1      =  ~  x^x-"*  X  mv^~^  =  ■—  mar*^!, 

\      ^  —  y       ly^x 

4.  When  7i  is  a  negative  fraction. 

Let  7i  =  — -,  in  which  p  and  q  are  supposed  to  be  positive 

( ^-t_  -I ) 

integers.    We  are  to  show  that    1 ^- —  (       =  —  ^a;~f  ~^* 

\     ^-y      )  v^x  q 


Q  —  y     q  _P     _P.lx^  —  yl\ 

.-Z —  =  -_  a;  qy  q\ -^  I 

x  —  y  ^      \x  —  yr 


X  g  — 

X 


2< 1/9   f  n    P—\ 

^—  f         =r^a;«       • 

^  ~-y     )  y=x      9. 


[x  1  —  y   i\  -P.  ^P.     p   t-i 

I ^~ —        =:  —X   qx   q  X-Xq        = 

\    X  —  y     ]y=^  q 


P  -^-1 

■^-X    q 


V    q 

Q 

549.  Let  us  now  find  the  expansion  of  (x  +  a)",  when  n  is 
positive  or  negative,  entire  or  fractional. 

X  -}-  a  =  x\l  +  -);    therefore     {x  +  «)"  =  a;" f  1  +  - j  . 

Hence,  the  expansion  of  {x  4-  a)^  may  be  obtained  by  multiply- 
ing that  of  (l  +  -)   by  x". 

Put  z=:^;  then  (l  +  ^)**=  (1  4-  «)*. 


BINOMIAL    FORMULA. 


359 


Assume   (1  +  0)^  =  A  +  B-2  +  Cz^  +  Dz^ -\~'Kz*  +  .  .  .  .  (1), 

in  which  A,  B,  C,  D,  E,  .  .  .  .  are  undetermined  coefficients  inde- 
pendent of  z. 

Suppose  0  =  0;  then  from  (1),  we  have  A  =  1. 

Substituting  1  for  A  in  (1),  we  have 


(1  +  2)»  =  1  H-  B;2  +  C22  -f  Bz^  +  E2*  +  . 


(2). 


Since  (2)  is  to  be  true  for  all  values  of  Zy  we  may  substitute 
any  letter  or  any  expression  for  z.  Substituting  u  for  z  in  (2), 
we  have 


(1  +  w)"  =  1  +  Bw  +  Cu^  +  DmS  +  Ew^  4- . 


(3). 


Subtracting  (3)  from  (2),  and  dividing  the  result  by  the  iden- 
tity  (l+z)--(l  +  n)=z-u, 


Now  suppose  u=z  z;    then,  by  the  Lemma,  (4)  becomes 


w  (1  +  z)»-^  =  B-\-2Cz+  Sm^  +  4E23  + 
Multiplying  (5)  by  1  +  2, 


(5), 


n{l-{-z)»=B  +  2C 
+  B 


z-\-3T) 

H-2C 


22  +  4E 

4- 3D 


(6). 


Multiplying  (2)  by  n, 
n(l-\-z)''=:n-\-7iBz-{-nCz^  +  nDz^-\-nEz^-{-. 
Equating  the  second  members  of  (6)  and  (7), 


•  •  •    •    • 


(7). 


B  +  2C> 

J  +  3D 

22  +  4E2: 

3+....=w4. 

nBz 

+  B 

+  2C 

+  3D 

B  = 

n 

2C  +    B  = 

nB 

•• 

3D  +  2C  = 

nQ 

4E  +  3D  = 

nD 

(8). 


Y, 


360 


SERIES. 


whence, 


B  =  n 

n(n  —  l) 

n(n—l){n—2) 
n{n—l)(n—2){n—3) 


C  = 
D  = 
E  = 


Substituting  these  values  in  (2),  we  have 
Substituting  for  z  its  value,  -,  (9)  becomes 


(9). 


(10). 


Multiplying  (10)  by  a", 

1*  .     12 

(11). 

Cor. — If  n  is  not  a  positive  integer,  the  expansion  of  {z  +  a)" 
is  an  infinite  series;  for  no  one  of  the  factors  in  the  coefficients 
can  be  equal  to  zero  under  this  hypothesis. 


j:xampz:es. 

Expand  each  of  the  following  expressions : 
1 


a  +  5 


2.     Vl  + 


a      or      a^      a^ 

Ans-l+la-la^  +  ^^a" 


SYKOPSIS    FOR    REVIEW.  361 

3.  (a-x)i.   ^ns,  ai(l-^-^,-^-^^,-.  ..  ), 

4.  (l-x)^.  ^^''  ^-3-3:3- 3^^^-"- 

5.  (a  +  J)t     ^^^^'-  «^  (1  +  3^ -3-7(3^ +3T6-.-9^3--- •)• 

1  ,        1       b       b^       b^       ¥ 

a  —  b  a      a^      a^      a*       w 

».     ^«     6j.  ^ws.  «   \^i      g^^       ^.^^2       3.6'9a3      •••7 

9.     (1— a)-3.  Ans.  l  +  3a  +  6a2  +  10«8+15a4+21tt«+.... 

^  .            ,   a;2        6a;3        6-11:^4 

^^'     Vf^'  ^^'•^+5+2^+2^3T«+--" 

550.  SYNOPSIS   FOR   REVIEW. 


CHAP.  XX. 
SERIES. 


General  Defini- 
tions. 


Amthmetical  Pro- 
gression. 


2'erms. 

Finite  Series. — Infinite  Series. 
^  Converging  Series. — JDicerg'g  Series. 

"  Extremes. — Common  difference. 

Increasing  A.  P. — Decreasing  A.  P, 

Notation. 

To  find  I,  when  a,  d,  and  n  are  given. 

To  find  s,  when  a,  I,  and  n  are  given. 

Sum  of  two  terms  equidistant  from 
extremes. 

To  insert  any  number  of  arithmeti- 
cal means  between  two  given 
quantities. 

To  find  any  two  of  the  quantities 
a,  I,  n,  d,  s,  when  the  three  others 
are  given. 

To  find  arith.  mean. 

Arith.  Mean,  i  To  find  a  and  I  when 
M,c?,  71,  are  given. 


SERIES. 


SYNOPSIS   FOR   BEVIEW^Cmiinved. 


CHAP.  XX. 

SERIES. 
Continued. 


Geometrical  Pro- 
gression. 


Treatm't  of  Series 

BY  THE  DlFFEREN- 

TiAii  Method. 


Development  of  Ex- 
press'n  into  Series. 


Extremes. — Ratio. 

Increasing  G.P.—Decreamng  O.  P. 

— Infinite  Decreasing  G.  P. 
Notation. 
To  find  I,  icTien  a,  r,  and  n  are 

given.    Cor. 
To  find    s,   when  a,  I,  and  r  are 

given.     Cor. 
Product  of  two  terms  equidistant 

from  extremes. 
To  insert  any  number  of  geometric 

means  'between  two  given  quan. 
To  find  the  continued  product  of  the 

terms  of  a  O.  P. 
To  find  any  two  of  the  quantities  a, 

I,  n,  r,  and  s,  when  the  three 

others  are  given. 
Geom  ^ '^o  find  geometrical  mean. 
Mean.  1  "^^  ^^  ^  ^^'^  ^  when  M,  r, 
'    and  »  are  given.  Cor.  1,2. 

'  Orders  of  differences. 
Differential  method. 
To  find  the  n""  term  of  a  series. 
To  find  the  sum  of  n  terms  of  a  scries. 
Interpolation.— FcfTmu\&  for  Inter- 
polation. 

'  Development  of  fractions  by  division. 
Development  of  expressions  of  the 

n  by  extract- 


form  of  "s/m  ± 

ing  the  indicated  root. 
Development  of  expressions  by  means 

of  Undetermined  Coefficients. 
Generating  Fraction. 
Scale  of  Relation. 
Order  of  Recurring  Series. 
To  find  the  Scale  of  Relation  in  a 

Recurring  Series.     Cor.  1,  2. 
To  find  the  Generating  Fraction, 

Sch. 
Reversion  op  Series. 

.  B1NOML&.L  Formula  for  any  Exponent,  -j  -5^^^^- 

{  Cor. 


Recurring  Series  . . 


OHAPTEE   XXI. 
LOGARITHMS   AND    EXPONENTIAL  EQUATIONS. 


LOGARITHMS, 


551.  The  Logarithm  of  a  number  is  the  exponent  by 
which  some  fixed  number  must  be  affected  in  order  to  produce 
the  given  number.  The  fixed  number  is  called  the  Hase  of  the 
System,  Thus,  in  the  equation  a''  =  ?^,  x  is  the  logarithm  of 
n  to  the  base  a. 

For  brevity,  the  expression  log^w  is  sometimes  used  to  denote 
the  logarithm  of  ?^  to  the  base  a»  Thus,  x  =  logaW  expresses  the 
same  relation  as  «*  =  n. 

552.  Any  number  except  +  1  and  -—  1  may  be  used  as 
the  base ;  hence  there  may  be  an  infinite  number  of  systems  of 
logarithms.  There  are  only  two  systems,  however,  in  general  use, 
namely:  Briggs'  system,  the  base  of  which  is  10,  and  Napier's 
system,  the  base  of  which  is  2.718  -f . 

Briggs'  system  of  logarithms  is  used  more  than  that  of  Napier, 
and  is  hence  called  the  common  system. 

553.  If  in  the  equation  a*  =  7i  we  suppose  n  to  represent  a 
perfect  power  of  «,  then  x  will  be  some  integer;  but  if  n  is  not  a 
perfect  power  of  a,  then  x  will  be  a  mixed  number  or  i\,  fraction. 

554.  The  Characteristic  of  a  logarithm  is  the  integral 
part  of  it,  and  the  3Iantissa  is  the  fractional  part.  Thus,  the 
characteristic  of   logg243    is    2,  and  the  mantissa  is  .5;    for 

92.8  _  9^^  35  _  243. 


364  LOGARITHMS. 


GENERAL    PROPERTIES    OF    LOGARITHMS. 

555.  In  any  system  the  logarithm  of  1  is  0. 
For  a*  =  1  when  x  =  0  (84,  CoE.). 

55^.  In  any  system  the  logarithm  of  the  base  is  1. 
For  a'^  =i  a  when  a;  =  1. 

551,  In  a  system  whose  base  is  greater  than  1,  the  logarithm 
ofOis  —  00 . 

For  «~*  =  -— ,   and   — z  =  0  when  «  >  1. 

558.  In  a  system  whose  base  is  less  than  1,  the  logarithm  of  0 
is  +  00 . 

For  a*  =:  0  when   a  <  1. 

559.  In  a  system  whose  base  is  positive , a  negative  quantity 
has  no  real  logarithm. 

For,  if  a  is  positive,  a*  is  positive,  whether  x  is  positive  or  neg- 
ative.   Thus,  102  =  100,  and  10-2  =  -L  =  ^. 

10^       100 

560.  The  logarithm  of  a  product  is  equal  to  the  sum  of  the 
logarithms  of  its  factors. 

Let  X  =  logflW,  and  y  =  log^w ; 

then  m  =  a^,  and  7i=za^; 

whence,  mn  =  a^^a^  =  a'^K 

Therefore,  ]og„7nn  =  x  ■{-  y  (551)  =  log^wi  +  hg^n. 

561.  TTie  logarithm  of  a  quotient  is  equal  to  the  remainder 
obtained  by  subtracting  the  logarithn  of  the  divisor  from  that  of 
the  dividend. 

Dividing  m  =  a'  by  n^^aM^ 

n 
hence  logo  —=  x  —  y  =  hg^m  —  log^w. 


LOGAKITHMS.  365 

563.  The  logarithm  of  any  poiuer  of  a  numher  is  equal  to  the 
product  of  the  exjjonent  of  the  power  and  the  logarithm  of  the 
number. 

Raising  both  members  of  the  equation  m  =  a*  to  the  r^^ 
power, 

loga(m^)  =  rX:=r  loga77Z. 

563.  TIte  logarithm  of  any  root  of  a  numher  is  equal  to  the 
quotient  obtained  by  dividing  the  logarithm  of  the  number  by  the 
index  of  the  root. 

Extracting  the  r^^  root  of  both  members  of  the  equation 
m  =  a*, 

,  {r /—\         X         logam 


5(54,  EXAMPLES. 

Prove  each  of  the  following  statements: 

1.  log  (abc)  =  log  a  +  log  b  +  logc. 

2.  log  \^J  =  log  a  +  log  5  4-  log  c  —  log  d. 

3.  log  {aWc^)  =  2  log  a  +  3  log  J  +  4  log  c. 

4.  log(^'-)  =  2]og«  +  31og5  +  41ogc- 5log^. 

5.  log  Vabc  =  ^  (log  a  +  log  J  +  log  c). 

6.  log  Va^  —  ^  =  I  [log  (a  +  b)  +  log  {a  —  b)]. 


S66  LOCAEITHMS. 


THE    COMMON    SYSTEM. 

565.   To  find  the  characteristic   of  a  logarithm  in 
the  common  system. 

In  this  system, 

log  100  =  log  1         =  0,  log  10-1  _  log  .1         =  —  1, 

log  101  ^  log  10        =  1,  log  10-2  =  log  .01        =  —  2, 

log  102  =  log  100      =  2,  log  10-3  =  log  .001      =  —  3, 

log  103  =  log  1000    =  3,  log  10-4  _  log  .0001    =  —  4, 

log  10^  =  log  10000  =  4,  log  10-s  =  log  .00001  =  —  5, 


Hence,  supposing  n  to  be  a  positive  integer, 

1st.  The  logarithm  of  a  number  between  10**  and  10"+^  is 
greater  than  n  and  less  than  n  -\-  1\  its  characteristic,  therefore, 
is  n.  Now,  the  number  of  figures  in  the  iutegi'al  part  of  a  num- 
ber between  10"  and  10"+*  is  n  +  1.     Hence, 

The  characteristic  of  the  common  logarithm  of  an  integer,  or 
of  a  number  composed  of  an  integer  and  a  decimal  fraction^  is  pos- 
itive and  one  less  than  the  number  of  figures  in  the  integral  part 
of  that  number.    Thus,  the  characteristic  of  logio  258.045  is  2. 


10-(»+i)j  that  is,  between  -—  and  -ttz-^x,  is  some  negative  number 


2d.  The  logarithm  of  a  decimal  fraction  between  10""  and 

10"»  ^^^  W 

between  —  n  and  —  (n  ■\-\)\  hence,  if  we  agree  that  the  man- 
tissa shall  in  all  cases  be  positive,  the  characteristic  will  be 
—  (7i  4- 1).  Now,  the  number  of  ciphers  preceding  the  first  sig- 
nificant figure  in  a  decimal  fraction  between  -—  and  — r^,  is  n. 
Hence, 

Tlie  characteristic  of  the  common  logarithm  of  a  decimal  frac- 
tion is  negative  and  numerically  one  greater  than  the  number  of 
ciphers  preceding  the  first  significant  figure  in  that  fraction. 
Thus,  the  characteristic  of  logio  .0546  is  —  2. 


LOGARITHMS.  367 

566.  If  the  ratio  of  two  numbers  is  any  perfect  power  of  10, 
the  mantissas  of  their  logarithms  in  the  common  system  will  he 
the  same. 

This  follows  from  Art.  561.  Thus,  denoting  the  mantissa  of 
logio  5468  by  m, 

log  5468  =  3  +  m, 

log  546.8  =  log  (^)  =  log  5468  —  loglO  =  3  -f  m  —1  —%+m, 

log  54.68  =  log  (^^)  =  log  5468  —  log  100  =  3  +m-2=l+m, 

log  5.468  =  log(^)  =  log  5468  -  log  1000=3 +  m-3=:0  +  m, 

log  .05468  =  log  (^-^^)  =  log  5468  -  log  100000  =  3+^-5 

=  —  2  +  m. 

COMPUTATION    OF    LOGARITHMS. 

567.  To  express  the  logarithm  of  a  number  in  terms 
of  that  number  and  the  base  of  the  system. 

Let  X  be  the  logarithm  of  n  to  the  base  a ;  then 

a'  =  n    ,    .    .    (1). 
Assume  a  =  1  -f-  t??,    and    n^l  -{- p\ 

then  (1)  becomeg 

{l+mY  =  l-^p    .    .    .    (2); 

whence,  (1 -f  m)*y=(lH-^)y  .    .    .    (3). 

Expanding  both  members  of  (3)  by  the  Binomial  Formula, 

=i+yp+y^-^f+y^y^-:^f+ (4). 


368  LOGARITHMS. 

Dropping  1  from  both  members  of  (4)  and  dividing  the  result 

x(xy  —  1)     -   ,   ir  (xy  —  1)  (xy  —  2)     _   . 

If  ^     l£ 

=^  +  (j^-iV  +  (y-iKv-^V+ (5). 

Making  y  =  0,  (5)  becomes 

m^         m^         7n*         w^ 


whence, 


^       2  +  3         4+5        ^^^' 


^  _  ^  a.  ^  _ -^  a.  ^  _ 

^        2+3         4  +  5        •  •  •  •  ,^, 

^  = 2 ^ 2 i^ •      •      •      (7). 

m^      m^      m*      m^  ' 

But   X  =  logaW  =:  loga  (1  ■\-  'p)\   hcncc,  if  we  put 

^^ 1 

rn?      m^      m^      m^  ' 

'"-y  +  y-T  +  T-----. 


we  have 

x  =  log„(l+^)=M(;,-^  +  |!-^V^-....)...(L). 

The  second  member  of  (L)  consists  of  two  factors,  tiamely : 
the  series  within  the  parenthesis,  which  depends  only  upon  the 
number,  and  the  quantity  M,  which  depends  only  upon  the  base 
of  the  system. 

The  factor  M,  which  depends  only  upon  the  base,  is  called  the 
modulus  of  the  system. 

The  series  in  (L)  is  called  the  Loffarithmic  Series, 

568.  To  find  the  Base  of  Napier's  System. 

Baron  Napier  arbitrarily  assumed  the  modulus  of  his  system 
to  be  unity.  Making  M  =  1,  and  denoting  the  Base  of  Napier's 
System  by  e,  (L)  becomes 


LOGARITHMS.  369 

^  =  log,(l+p)=p-f  +  f-^  +  f- (1). 

Eeverting  the  series  in  the  second  member  of  (1),  we  obtain 

X^  CC^  X^    ,      0^    ,      Ofi     ,  /Qx 

^=^+l-+l+l+l+f  + (^^• 

But  6^  =  1 +i?; 

x^       a^       x^       x^       x^  ,o\ 

Making  a;  =  1  in  (3),  we  have 

1111.1.  ^A\ 

^  =  ^  +  l2+f +  f  +  f+^+ (^)- 

Summing  the  series  in  (4)  to  nine  terms,  we  find 

e  =  2.718282. 

569.  T7ie  logarithm  of  a  number  in  any  system  is  equal  to 
the  product  obtained  by  multiplying  the  modulus  of  that  system  by 
the  Napierian  logarithm  of  the  same  number. 

For    log„(l  +^)  =  M  (^  -  I  +  I-  1^  +  ^-. . .  .)(567), 
and      \os,(i+p)=p-^^l._t  +  ^_  ....  (568); 


C4.1A«J.                  iV^g    ^J. 

T-i^y-y^       2^3         4^5 

.*. 

loga(l+i?)=Mloge(l+i?), 

Cor.— If 

l^p  =  a, 

we  have 

\ogaa  =  Clogs', 

but 

loga^  =  1  (556) ; 

•*• 

1  =  M  logeffi ; 

whence, 
Hence, 

l0ge« 

The  modulus  of  any  system  is  equal  to  the  reciprocal  of  the 
Napieria7i  logarithm  of  the  base  of  the' system. 
U 


370  LOGARITHMS. 

570.  To  transform  the   Logarithmic   Series  into  a 
Converging  Series. 

The  formula 
log„(l+i,)  =  M(;,_^  +  ^-^  +  |^-....)    .   .   .  (1) 

cannot  be  used  for  the  computation  of  logarithms  when  ^  >  1, 
because  the  series  in  its  second  member  does  not  converge. 
Substituting  —p  for  jt?  in  (1),  we  have 

log.(l-,)  =  M(-,-|-|_^-^-. ...)...(.). 

Subtracting  (2)   from   (1),  observing    that    loga(l-fj9)  — 
log«(l-i<)=log,{i±-|)(661), 

^^^•([^|)  =  ^^K^+l  +  l  +  ^+--)    •    •    •     (^)- 

.  1  .,         1+jy      z  +  1 

Assume  p  =  t: — -— r ;    then -■  = . 

^       2^;  +  1  1  —p         z 

Substituting  these  values  in  (3), 

l0g«  y-^)  —  ^^oa  (2;  +  1)  —  l0ga2;  = 

^^^(rin[  + 3(2^3  + 5(^^  •  •  •  (^>- 

For  Napier's  System  (4)  becomes 
loge  (z  +  1)  —  logeZ  = 

^  (2^^  "^  3  (2^  +  1)3  +  5  (2;^  +  If  + )     •    •    •     •     (^)' 

whence,  by  transposition, 
log,(^  +  l)  = 

^^^^  +  ^27^1  + 3^2^3+57^^  •  •  •  W- 


LOGARITHMS.  371 

571.  To  compute  a  Table  of  Napierian  Logarithms. 

log,0  =  -  00  (551), 
log,l  =  0  (555), 

log,2  =  log.l  +  2(3U3i^3+^  +  ^+....) 
=  0.693147  (570), 

log.3=.log.24-2(l-|-3L_^+^  +  y^+....) 

=  1.098612, 
log,4  =  log,22  =  2  log,2  =  1.386294, 

log,5  =  log,4  +  2g  +  3i^3  +  ^,  +  ^+  ....) 

=  1.609438, 
log,6  =  log,2  +  log,3  =  1.791759, 

log.7  =  log/.  +  2(l+3434-^3+  ....)  =  1.945910, 

log,8  =  loge23  =  3  loge2  =  2.079442, 
log,9  =  log,32  =  2  log,3  =  2.197225, 
logelO  =  log,2  4-  logeS  =  2.302585, 

572.  To  find  the  modulus  of  the  common  system. 
Denoting  the  modulus  of  the  common  system  by  M,  we  have 

573.  To  compute  a  table  of  common  logarithms. 

If  we  multiply  the  Napierian  logarithm  of  a  number  by  the 
modulus  of  the  common  system,  the  product  will  be  the  common 
logarithm  of  the  same  number.    Thus, 

logio5  =  1.609438  x  .434294  =  0.698970. 


373 


LOGAKITHMS. 


TABLE   OF  COMMON"   LOGARITHMS  FROM   1  TO   100. 


N. 

Loo. 

N. 

Loo. 

N. 

Loo. 

N. 

Loo. 

1 

0.000000 

26 

1.414973 

51 

1.707570 

76 

1.880814 

2 

0.301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0.477121 

28 

1.447158 

53 

1.724276 

78 

1.892095 

4 

0.602060 

29 

1.462398 

54 

1.732394 

79 

1.897627 

5 

0.698970 

30 

1.477121 

55 

1.740363 

80 

1.903090 

G 

0.778151 

31 

1.491362 

56 

1.748188 

81 

1.908485 

7 

0.845098 

32 

1.505150 

57 

1.755875 

82 

1.913814 

8 

0.903090 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

0.954243 

34 

1.531479 

59 

1.770852 

84 

1.924279 

10 

1.000000 

35 

1.544068 

60 

1.778151 

85 

1.929419 

11 

1.041393 

36 

1.556303 

61 

1.785330 

86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519 

13 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

39 

1.591065 

64 

1.806180 

89 

1.949390 

15 

1.176091 

40 

1.602060 

65 

1.812913 

90 

1.954243 

16 

1.204120 

41 

1.612784 

66 

1.819544 

91 

1.959041 

17 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.643453 

69 

1.838849 

94 

1.973128 

20 

1.301030 

45 

1.653213 

70 

1.845098 

95 

1.977724 

21 

1.322219 

46 

1.662758 

71 

1.851258 

96 

1.982271 

22 

1.342423 

47 

1.672098 

72 

1.857333 

97 

1.986772 

23 

1.361728 

48 

1.681241 

73 

1.863323 

98 

1.991226 

24 

1.380211 

49 

1.690196 

74 

1.869232 

99 

1.995035 

25 

1.397940 

50 

1.698970 

75 

J 1 

1.875061 

100 

2.000000 

574,  EXAMrT.ES. 

1.  Find  the  product  of  9  and  7  by  means  of  logarithms. 
Log  (9  X  7)=:log9+log  7  (560)  =0.954243  +  0.845098=1.799341. 
The  number  corresponding  to  this  logarithm  is  63  (573). 


EXPONENTIAL    EQUATIONS.  373 

2.  Divide  210  by  7  by  means  of  logarithms. 

Log  (^i^)=log210-log  7=2.322219-0.845098=1.477121 
=  log  30. 

3.  Find  the  square  of  9  by  means  of  logarithms. 

4.  Find  the  fourth  root  of  625  by  means  of  logarithms. 

5.  Find  the  logarithm  of  33^. 

6.  Find  the  logarithm  of  6^  x  7^  x  8^. 

EXPONENTIAL    EQUATIONS. 

575.  An  Exj^onential  Equation  is  one  in  which  the 
unknown  quantity  occurs  as  an  exponent.  Thus,  a^  =  n  is  an 
exponential  equation. 

576.  To  solve  the  exponential  equation    a^  =  n. 

Taking  the  logarithm  of  each  member  of  this  equation,  we 
have 

xloga  =  log  71  (562) ; 

whence,  x  =  ——. 

log  a 

EXAMPTjES. 

Solve  each  of  the  following  equations : 

1.  3^  =  27.  Ans.  x  =  3. 

2.  5^  =  100.  Ans.  x  =  2.861. 

Ans.  1.5. 
Ans.x  =  ^.^^^P^. 

A  log  J 

Ans.  x—  ^ 

6.  a^  —  2pa^  =  I,  Ans.  x 


3. 

3 

2^  = 

=  4. 

4. 

ah^ 

=  n. 

5. 

1 

=  n. 

log  n  —  log  a 

\og{p±Vb+p^) 
log  a 


374 


LOQABITHMS    ±ND    EXPONENTIAL    EQUATIONS. 


5111. 


SYNOPSIS   FOR   REVIEW. 


LOGARITHMS. 


Log.  of  a  number. 
Base  of  a  system. 
Characteristic. — Mantissa. 


General 
Properties. 


Common 

System. 


Computation.  . 


Log.  1. 
Log.  Base. 
When  Base  >•  1. 
WTien  Base  <  1. 
When  Base  is  positive. 
Log.  of  a  Product. 
Log.  of  a  Quotient. 
Log.  of  a  Power. 
L  Log.  of  a  Root. 

To  find  th^  characteristic. 
Mantissas  of  log.  of  two  ?ium- 

bers  whose  ratio  is  a  perfect 

power  of  10. 

""  To  express  log.  of  a  number  in 
terms  of  that  number  and  the 
base  of  the  system. 

Modulus. — Logarithmic  series. 

To  find  the  base  of  Napier's  sys- 
tem. 

Log.  number  =  Modulus  x  Na- 
pier's Log.  same  number. 

Modulus  of  any  system  =  recip- 
rocal of  the  Napi.rian  log.  of 
la83  of  the  system. 

To  transform  log.  scries  into 
converging  series. 

To  compute  table  of  Napierian 
log. 
^  To  compute  table  of  common  log. 


I  EXPONENTIAL  EQUATIONS. 


CHAPTEE   XXII. 
COMPOUND    INTEREST    AND    ANNUITIES. 


COMPOUND    INTEREST, 


578.  To  find  the  amount  of  p  dollars  at  compound 
interest  for  n  years   at  r  per  cent,  per  annum. 

At  tlie  end  of  the  1st  year  the  amount  will  be 

p  -irpr=p{l  +  r); 

at  the  end  of  the  2d  year  the  amount  will  be 

p{l  +  r)  +  p{l  +  r)r  =p{l  +  r  )2; 

at  the  end  of  the  3d  year  the  amount  will  be 

^  (1  +  r)2  +i?  (1  +  rfr  =p  (1  +  rf, 

and  so  on.    Hence,  denoting  the  requhed  amount  by  A, 

A=i?(l+r)»    .    .    .    (1). 

Any  one  of  the  four  quantities,  A,  p,  n,  and  r  may  be  found 
from  this  equation  when  the  three  others  are  given.  The  compu- 
tation is  most  readily  performed  by  means  of  logarithms.  Taking 
the  logarithm  of  each  member  of  (1), 

log  A  =  log;?  +  ^  log  (1  +  r)    .    .    .     (2)  (560-562) ; 
whence,     log;;  =  log  A  —  w  log  (1  +  r)     .     .    .     (3), 

log(l  +  r)z.^"g^-'°g-P    .    . 


n 


(4), 


_logA  —  log;? 


and  n  =  -:p—j——~     .     .     .     (5). 

log  (1  +  r)  ^  ^ 


37G  ANNUITIES. 

Example. — How  much  will  $500  amount  to  in  five  years  at  6 
per  cent,  compound  interest? 

Given    jl^gl-^^      =0.025306) 
( log  609.10  =  2.825491  5  * 

Substituting  500  for  j9,  .06  for  r,  and  5  for  n  in  (2),  we  have 

log  A  =  log  500  +  5  log  1.06 

=  2.698970  +  5  X  .025306  =  2.825500. 

Since  log  669.10  =  2.825491,  it  fx)llows  that  A  =  $669.10. 

ANNUITIES. 

579.  An  Annuity  is  a  sum  of  money  which  is  payable 
annually.  The  term  is  also  applied  to  a  sum  of  money  payable  at 
any  equal  intervals  of  time. 

580.  To  find  the  amonnt  of  an  annuity  of  a  dollars 
for  n  years  at  r  per  cent,  per  annum,  when  the  inter- 
est is  compounded  every  year. 

The  first  payment  a  becomes  due  at  the  end  of  the  first  year, 
and  m  n  —  1  years  this  will  amount  to  «  (1  -f  r)"~i  (578) ;  the 
second  payment  a  becomes  due  at  the  end  of  the  second  year,  and 
in  n  —  2  years  this  will  amount  to  « (1  +  r)«-2j  the  third  pay- 
ment will  amount  to  a  (1  +  r)"-^  in  n  —  Z  years ;  and  so  on. 
Hence,  denoting  the  amount  of  the  annuity  by  A, 

A  =  a  (1  +  r)"-i  +  «  (1  +  r)"-2  +  a  (1  -f  r)"-3  +  .  .  .  . 
+  « (1  +  r)  +  «    .     .    .     (1). 

By  reversing  the  order  of  the  terms  in  the  second  member  of 

(1). 

A=a  +  f((l  +  r)  +  a(l  +  r)2+.---  +«(!  +  '•)"-'  ■  •  •  (3); 
whence,  A  ^    (1  +  ,.)  _T- "=  " '  r "    '    '     ^^^- 


SYNOPSIS    FOR    REVIEW. 


377 


581.  To  find  the  present  value  of  an  annuity  of  a 
dollars  for  n  years,  at  f  per  cent,  per  annum,  the  in- 
terest being  compounded  every  year. 

Denoting  the  present  value  of  the  annuity  by  P, 

(1  4-  r)»  - 1 


P(l+r)' 


whence, 


P=: 


a    (1  +  r)«  —  1 


(1  +  tY 
Cor. — If  ?^  =  00 ,    (2)  becomes 
P  =  ^. 


(1)  (578-580) ; 
(2). 


^^2,. 


SYNOPSIS    FOR    REVIEW. 


r  ooMPouro 

INTEREST 


CHAP.  XXII. 


M 


To  FIND  THE  AMOrifT  OP  ^  DOLLARS 
AT  COMPOUND  INTEREST  FOR  71  YEARS 
AT  r  PER  CENT.    PER  ANNUM. 


AOTTJITIES. 


^  To  FIND  THE  AMOUNT  OF  AN  ANNUITY 
OF  a  DOLLARS  FOR  71  YEARS  AT  T  PER 
CENT.  PER  ANNUM,  WHEN  THE  INTER- 
EST IS  COMPOUNDED  EVERY  YEAR. 
To  FIND  THE  PRESENT  VALUE  OF  AN 
ANNUITY  OF  a  DOLLARS  FOR  7^  YEARS 
AT  r  PER  CENT.  PER  ANNUM,  WHEN 
THE  INTEREST  IS  COMPOUNDED  EVERY 
YEAR.  Cor. 


CHAPTEE    XXIII. 
THEORY    OF    EQUATIOIsTS. 


DEFINITIONS. 


583.  Every  equation  of  the  n^  degree  containing  only  one 
unknown  quantity  may  be  written  under  the  form  of 

a;"  +  Ao:"-!  +  Bs^-^  +  ....+  Ka;  +  L  =  0. 

This  equation  is  called  the  general  equation  of  the  n^^  degree. 
The  term  L,  which  is  called  the  absolute  or  independent  term,  may 
be  considered  as  the  coefficient  of  aP. 

584.  A  Function  of  a  quantity  is  an  expression  contain- 
ing that  quantity.     Thus,   aoi?  -{-  bx    is  a  function  of  x. 

For  brevity  we  shall  sometimes  use  the  symbol /(ic)  to  denote 
&  function  of  x. 

If  f(x)  is  entire  and  rational  with  reference  to  a;,  it  is  called 
a  rational  integral  function  of  x. 

In  the  present  Chapter,  when  f{x)  is  used  without  modifica- 
tion, it  is  understood  to  denote  a  rational  integral  function  of  x. 

585.  Any  quantity,  which  substituted  for  x  in  f  (x)  causes 
f(x)  to  vanish,  is  a  Moot  of  the  equation  f{x)  =  0. 

GENERAL    PROPERTIES. 

586.  If  f{x)  vanishes  when  x=^r,  the  function  is  divisi- 
ble by  X  —  r. 

Suppose  f{x)  to  be  divided  by  x  —  r,  and  the  operation 
continued  until  a  remainder  is  obtained  which  is  independent  of  x. 


GEN-ERAL    PROPERTIES.  379 

Denote  the  quotient  by  Q  and  the  remainder,  if  there  be  one,  by  R; 
we  then  have  the  identity 

f(x)=Q{x-r)-{-K 

By  hypothesis,  / (x)  vanishes  when  x  =  r;  and  since  Q  is  a 
rational  integral  function  of  x,  it  cannot  become  infinite  when 
x  =  r;  hence  Q{x  —  r)  vanishes  when  x=r.  Therefore  E 
vanishes  when  x  =  r.  But  R  does  not  contain  x',  hence  it  van 
ishes  without  regard  to  the  value  of  x. 

^SH.  If  f{x)  is  divisible  hy  x  —  r,  then  r  is  a  root  of  the 
equation  f{x)  =  0. 

Let  Q  denote  the  quotient  obtained  by  dividing  f{x)  by 
x  —  r;    we  shall  then  have  the  identity 

f{x)  =  (i{x-r). 

Now  Q{x  —  r)  vanishes  when  x  =  r;  hence  f(x)  vanishes 
when  X  =  r.  It  therefore  follows  that  r  is  a  root  of  the  equation 
f{x)=0  {5S5). 

588.  If  the  equation  f{x)  =  0  is  of  the  n^  degree,  it  has  n 
roots,  and  no  more. 

Let  a  represent  a  root  of  the  equation 

f{x)=0    .    .    .     (1); 

then  f{x)  is  divisible  hjx  —  a  (586).  The  quotient  obtained 
by  dividing  f(x)  hy  x  —  a  will  be  of  the  (n  —  iy^  degree. 
Denoting  the  quotient  by  fi(x)y  (1)  may  be  written 

{x-a)f,(x)=0    .    .    .     (2). 

Again,  let  b  represent  a  root  of  the  equation 

Mx)=0    .    .    .    (3), 

which  is  of  the  (n  —  iy^  degree;  then  f^ix)  is  divisible  by 
x  —  b.  The  quotient  obtained  by  dividing  /i  (a;)  hj  x  —  b  will 
be  of  the  {n  —  2y^  degree.  Denoting  this  quotient  by  f^ix), 
(2)  may  be  written 

{x-a){x-h)f,{x)  =  0    .    .    .     (4). 


380  THEORY    OF   equatio:n"S. 

By  continuing  this  process,  f{x)  will  ultimately  be  resolved 
into  n  binomial  factors,  x—a,x—h,  x—CyX—d,  .  .  .  .,  (x—k), 
x—l. 

r.  f(x)  =  {x-a)  {x-b)  (x-c)  (x-d) ....  (x-k)  (x-l)  . .  .  (5). 

Now  f{x)  vanishes  when  x  is  equal  to  any  one  of  the  n  quan- 
tities a,  b,  c,  df  .  .  .  .  Tc,  l\  hence  f{x)  =  0  has  n  roots.  This 
equation  has  no  more  than  n  roots,  for  if  we  ascribe  to  a;  a  value 
m  which  is  not  one  of  the  n  values  a,  b,  c,  d, .  .  , .  h,  I,  the  value 
of  / {x)  becomes  {m —a){m—  b) {m ~c)(m—d).,.  (m — k) {m — /) , 
which  is  not  zero,  because  each  factor  is  different  from  zero. 

Cor. — K  a  is  a  root  of  the  equation  f(x)  =  0,  then 
f{x)=z{x  —  (f)ft{^),  where  ft{x)  is  one  degree  lower  than 
f{x) ;  hence  the  remaining  roots  of  the  equation  f{x)  =  0  can 
be  found  if  we  can  solve  the  equation  f^  (x)  =  0.  In  like  man- 
ner, if  a  and  b  are  roots  of  the  equation  /  {x)  =  0,  then 
f{x)  =  [x — a)  (x — b)f^{x)  ;  hence  the  remaining  roots  of  the 
equation  f{x)  =0  can  be  found  if  we  can  solve  the  equation 
/jj  {x)  =  0,  which  is  .  two  degrees  lower  than  the  equation 
/{x)  =  0. 

589.  To  find  an  eqiiation  when  its  roots  are  given. 
Let  a,  b,  c,  df . .  .  .  k  be  the  n  roots  of  an  equation ;  then 

(x—a)  (x—b)  (x—c)  (x—d)  ....  (x—k)  =  0 

will   be  the  equation  required;    for  each  of  the  n  quantities, 
a,  bf  c,d,....  k  is  a  root  of  this  equation,  and  it  has  no  other  roots. 

MULTIPLICATION"  BY   DETACHED   COEFFICIENTS. 

590.  To  multiply  a  rational  integral  function  of  x 
by  X  ±_a,  by  means  of  detached  coefficients. 

JEXAMPZBS. 

1.  Multiply  a^  -\-  ox^  —  Gx  +  4:  hj  X  —  d. 
Since  the  coefficients  of  the  product  do  not  depend  upon  x, 
the  product  may  be  found  as  follows : 


GEXEBAL    PROPERTIES.  381 

1  +  5  —    6  +    4  Detached  coefficients  of  multiplicand. 

1  —  3 "  «  «  multiplier. 

1  +  5  —    6  +    4  Detached  coefficients  of  1st  partial  product. 

_  3  _  15  -^-  18  — 12         «  "         "  2d      "  " 

1+2  —  21  +  22  —  12  Detached  coefficients  of  product. 

Hence  the  product  is  a^-\-2a:^  —  21x^  +  22^;  —  12. 

Since  the  coefficients  of  the  first  partial  product  are  identical 
with  those  of  the  multiplicand,  this  operation  may  be  still  further 
abridged  as  follows : 

1  +  5—  6+  4         Detached  coefficients  of  multiplicand. 
_3_15  +  18_12 


1  +  2—21  +  22—12        «  "  "product. 

Multiplying  1,  the  coefficient  of  the  first  term  of  the  multi- 
plicand, by —  3,  and  adding  the  product  to  5,  we  obtain  2;  multi- 
plying 5  by  —  3,  and  adding  the  product  to  —  6,  we  obtain 
—  21 ;  multiplying  —  6  by  —  3,  and  adding  the  product  to  4, 
we  obtain  22 ;  and  multiplying  4  by  —  3,  we  obtain  —  12. 

When  multipUcation  is  performed  in  this  way,  the  terms  should 
be  arranged  according  to  the  powers  of  x ;  and  if  a  term  is  want- 
ing, its  place  should  be  ffiled  with  a  cipher. 

2.  Multiply    7^-\-Qx^  ^  5a;  —  10  by  a;  —  5. 

5   1  +  0  +  6+    5  —  10 

__  5  _.  0  —  30  —  25  +  50 


1  _  5  +  6  —  25  —  35  +  50 
Hence  the  product  is  a;^  —  5a;^  +  Q>x^  —  %bx^  —  35a;  +  50. 

3.  Multiply  2:5  _  4^  ^  6a:2  —  8a;  +  15    ^y  a:  +  8. 


8 


1  +  0-4+6—8  +  15 
+  8  +  0  —  32  +  48  —  64  +  120 


1  +  8  —  4  —  26  +  40  —  49  +  120 
Product,    a;6  +  8a;5  —  4a;4  —  26a;3  _|_  40a;2  _  4Qx  +  120. 

4.  Multiply   a;7  —  4ar3  +  6a;  —  7  by  a;  +  3. 

Ans.  x^  +  3a;7  —  4a;*  —  12a;3  +  6a?J  +  11a;  —  21. 


382  THEORY    OF    EQUATIOi^S. 

DIVISION  BY  DETACHED  COEFFICIENTS. 

591.  To  divide  a  rational  integral  function  of  x  by 
X  ±  «j  ^1  means  of  detached  coefficients. 

EXAMPIjEa. 

1.  Divide  a^s  —  9a:2  ^  26a:  —  24   by  x  —  4. 
Since  the  coefficients  of  the  quotient  do  not  depend  upon  Xy 
the  quotient  may  be  found  as  follows: 

Coeflfic'ts  of  dividend,  1—9  +  26— 24[1— 4  Coefficients  of  divisor. 
1—4  1—5  +  6      "        "quotient. 

—5  +  26 
—5  +  20 

6—24 
6-24 

Hence  the  quotient  is  ^  —  hx-\-^. 

This  operation  may  be  still  further  abridged  as  follows : 

1-9  +  26-24    -4    .    .    .    (A). 
_  4  +  20  —  24 


1  — 5  4-    G  +    0 

The  coefficient  of  the  first  term  of  the  quotient  is  evidently  1 
Multiplying  1,  the  first  coefficient  in  the  dividend,  by  —  4,  and 
subtracting  the  product  from  —  9,  we  obtain  —  5,  which  is  the 
second  coefficient  in  the  quotient ;  multiplying  —  5  by  —  4,  and 
subtracting  the  product  from  26,  we  obtain  6,  which  is  the  third 
coefficient  in  the  quotient ;  and  multiplying  6  by  —  4,  and  sub- 
tracting the  product  from  —  24,  we  obtain  0. 

"We  may  substitute  addition  for  subtraction  in  (A),  if  we  mul- 
tiply by  +  4 ;  thus, 

l_-94.26-24   4    .    .    .     (B). 
1_5-|.   6+  0 


GEN-ERAL    PROPERTIES.  383 

When  division  is  performed  by  means  of  detached  coefficients, 
the  terms  should  be  arranged  according  to  the  powers  of  x ;  and 
if  a  term  is  wanting,  its  place  should  be  filled  with  a  cipher. 

The  process  used  in  (B)  is  called  Synthetic  Division. 

2.  Divide  x^  —  ^a?  —  Ibx^  +  49a;  —  12  by  sr  —  5. 

l_3_15_j.49—  12 
4.54-10—25  +  120 


1  +  2—  5  +  24  +  108 
Hence  the  quotient  is  a:^  4.  ^^^  __  5^;  4.  24,   and  the  remain- 
der is  108. 

3.  Divide  x*  —  Sx^  —  \lx^  +  198a;  —  360  by  cc  —  7. 

1  _  8  —  11  +  198  —  360  1  7 

4  7  —    7  —  126  +  504  1 
1  _  1  _  18  +    72  +  144 

Hence  the  quotient  is  o?  —  a?  —  IHx  -\-  T2,  and  the  remain- 
der is  144. 

4.  Divide  o^  +  bx^  +  ^x  —  %  by  a;  +  4. 


1  +  5  +  2  —  8 
_4_4+8 


—  4 


1 +l_2+0 
Hence  the  quotient  is  a;2  +  a;  —  2. 

593.  GENEBAZ    EXAMPLES, 

1.  Show  that  1  is  a  root  of  the  equation 

a;3  4.  3a:2  _  iq^  +  12  =  0. 

That  the  first  member  of  this  equation  is  divisible  by  a;  —  1 


may  be  proved  as  follows : 


1  +  3-16  +  12 
414    4-12 


1(591); 


1  4  4  —  12  +    0 
hence  1  is  a  root  (587). 


384 


THEOEY    OF    EQUATIONS. 


2.  Show  that  3  is  a  root  of  the  equation  t^  —  lOa^  +  35a.'2 
—  50a;  +  24  =  0. 


3.  Show  that 
+  12a;  +  35  =  0. 


7  is  a  root  of  the  equation  a;*  +  22;^  —  Z\x^ 


4.  Show  that  —  1  and  —  2  are  roots  of  the  equation  a;^  —  4a;^ 
+  22a;3  _  25a;  -  42  =  0. 


5.  Show  that  1  +  V—  5  and  1  —  \/- 
equation  a;*  _  2a;8  ^  ^^  ^  10a;  —  30  =  0. 


5  are  roots  of  the 


6.  One  root  of  the  equation    a;^  +  5a;2  _|_  2a;  _  8  =  0  is  1 ; 
what  are  the  other  roots? 

1+5+2-8 
+1+6+8 


1-5 
—  1 

_  7-1-29  +  30 
+    6  +    1—30 

1  —  6 
—  2 

_  1  +  30  +  0 
+  16-30 

1+6+8+0 
a^J  +  Ga;  +  8  =  0  {^^^^  CoR.) ;    whence,    a;  =  —  2  or  —  4. 

7.  Two  roots  of  the  equation  a;*— 5a;3— 7a;2^29a;+30=0    are 
—  1   and   —  2 :    what  are  the  other  roots  ? 

—  1 

—  2 

1_8    +15+    0 
a;2  _  8a;  +  15  =  0 ;    whence,    a;  =  3   or  5. 

8.  Three  roots  of  the  equation  a:5_4^^22a;2— 25a;— 42  =0 
are   —  1,    —  2,  3 ;   what  are  the  other  roots  ? 

Ans.  2  +  V^^,  2  —  V"^^. 

9.  Two  roots  of  the  equation  7^  —  Zx^  —  \x^  -\-  30a;  —  36  =  0 
are  2  and  —  3 ;  what  are  the  other  roots  ? 

Ans.  2  +  V^^,   2  —  V^^. 

10.  One  root  of  the  equation    7?  —  \  =  ^    is  1 ;  what  are  the 
other  roots?  Ans.  i(—  1  ±  a/^^). 


2 

1  —1 

+  2 

-2 

4 

1  +1 

+  4 

—  2 

+  4 

-8 

GEITERAL    PKOPERTIES.  385 

11.  Find  the  equation  whose  roots  are  1,   —  2,   —  4. 
The  required  equation  is 

{x^l)[x-{-2)][x-{-4.)]  =  {x-l){x  +  2){x  +  ^)=:0. 
The  indicated  multiplication  may  be  performed  as  follows: 


(590); 


1+5     -1-2—8 

hence  a^  -\-  bx"^  -\-  2x  —  S  =  0    is  the  required  equation 

in  its  simplest  form. 

12.  Find  the  eouation  whose  roots  are  3,    —  2,    —  1,  5. 

Ans.  x^  —  6x^  —  W  +  2dx-{-dO=  0. 

13.  Find  the  equation  whose  roots  are  1  +  V"^,  1  — V— 5, 
//5,    __  //s.  Ans.  x^—2a^-{-x^-{-  10a;  —  30  =  0. 

14.  Find  the  equation  whose  roots  are  —1,  —2,  3,  2  + V— 3, 
2  _  V^Is.  Ans.  x^  —  4:X^  +  22:^2  _  25a;  —  42  =  0. 

15.  Find  the  equation  whose  roots  are  a,  b,  c. 

Ans.  7^  —  {a  ■\-  b  -\-  c)  x^  -\-  (ab  +  ac  -{-  bc)x—  abc  =  0. 

16.  Find  the  equation  whose  roots  are  a,  b,  c,  d. 

Ans.  x!^—(a-\-b-\-c  +  d)x^-\-{ab-\-ac-\-ad-\-bc-\-bd-\-cd)a? 
■^  (abc -^abd-\-acd-\-bcd)x-\- abed  :=0. 

593.  To  find  the  relation  between  the  coefficients 
off{x)  and  the  roots  of  the  equation  /  (x)  =0. 

Suppose  the  terms  of  f(x)  to  be  arranged  according  to  the 
descending  powers  of  x  and  that  the  coefficient  of  the  first  term  is 
1;  then 

1.  The  coefficient  of  the  second  term  with  its  sign  changed  is 
equal  to  the  sum  of  the  roots  (593,  15,  16) ; 

2.  The  coefficient  of  the  third  term  is  equal  to  the  sum  of  the 
products  of  the  roots,  taken  two  and  two  j 

25 


886  THEORY    OF    EQUATIONS. 

3.  TJie  coefficient  of  the  fourth  term  with  its  sign  changed  is 
equal  to  the  sum  of  the  products  of  the  rootSytahen  three  and  three; 
and  80  on. 

4.  If  the  degree  of  the  equation  is  even^  the  absolute  term  is 
equal  to  the  product  of  all  the  roots.  If  the  degree  of  the  cquaimi 
is  odd,  the  absolute  term  with  its  sign  changed  is  equal  to  the 
product  of  all  the  roots. 

By  a  method  similar  to  that  employed  in  Art.  472  it  may  be 
proved  that  these  laws  are  true  universally. 

Cor.  1. — If  the  roots  of  f(x)  =  0  are  all  negative,  each  term 
of  f{x)   is  positive. 

Cob.  2. — If  the  roots  of  f(x)  =  0  are  all  positive,  the  signs 
of  the  terras  of /(a;)  will  be  alternately  +  and  — . 

Cor.  3. — If  the  second  term  of  f{x)  does  not  appear,  the  sum 
of  the  roots  of  the  equation  /  (x)  =  0  is  equal  to  zero.  Thus, 
the  sum  of  the  roots  of  tlie  equation  a;^  —  2a;  -f  4  =  0  is  zero. 

Cor.  4. — K  f(x)  has  no  absolute  term,  at  least  one  of  the 
roots  of  f(x)  =0  is  zero.  Thus,  one  root  of  the  equation 
a^  —  2ci^  +  3x  =  0ia0. 

Cor.  5. — The  absolute  term  of  f{x)  is  divisible  by  each  root 
of  the  equation  f{x)  =  0. 

Cor.  6. — Let  a,  b,  c,  d, .  .  .  .  I  denote  the  roots  of  the  equa- 
tion a^+Aa;"-i  +  Ba:«-?+ 4-Ka:  +  L  =  0;   then 

—  A  =  «  +  Z>  +  c  +  6/+ +  I, 

and  B^ab  -\-  ac  +  ....-{-  bd  +  be -^  . . . . ; 

whence,      A^  —  2B  =  a^  -}- b^ -{- c^  +  d^ -\- +  P; 

that  is,  A2  —  2B  is  equal  to  the  sum  of  the  squares  of  the  roots 
of  the  proposed  equation.  Hence,  if  A^  —  2B  is  negative,  the 
roots  of  the  equation  cannot  be  all  real.  Thus,  the  roots  of  the 
equation  a^  —  4:X^  -\-  22a^  —  25x  —  42  =  0  are  not  all  real,  for 
(—4)2  —  2x22    is  negative. 


GEKEEAL    PROPERTIES.  '38? 

594.  An  equation  ivhose  coefficie7its  are  integers,  that  of  its 
first  term  being  unity,  cannot  have  a  root  which  is  a  rational 
fraction. 

Let  the  equation  be 

x^  +  A:k^-i  +  Ba:«-2  +....4-Ka;  +  L  =  0    .    .     .     (1), 

in  which  the  coefficients  A,  B,  .  .  .  .  K,  L  are  supposed  to  be  in- 
tegers. 

Suppose,  if  possible,  that  (1)  has  a  rational  fractional  root 

which  in  its  lowest  terms  is  expressed  by  t-     Substituting  t  for  x 

in  (1),  and  multiplying  the  resulting  equation  by  h"~\  wq  obtain 

J  +  Aa«-i  +  Ba«-2^>  +  ....  4-  KaZ>"-2  +  L2»«-i  =  0  .  .  .    (2) ; 

whence, 

j  =  -  (Aa"-i  +  B«"-2J  +....+  K«5«-2  +  LJ^-i)  .  .   .   (3). 

The  second  member  of  (3)  is  an  integer,  and  its  first  member 
is  an  irreducible  fraction.  Hence  j-  cannot  be  a  root  of  the  pro- 
posed equation. 

595.  If  a-^bV^  1  is  a  root  of  an  equation  whose  coeffi- 
cients are  real,  then  loill  a  —  bs/—\   be  a  root  of  that  equation. 

Let  a  -\-  b^/  —1   be  a  root  of  the  equation 
^n  _|.  ^^n-1  j^  Ba;"-2  +  .  .  .  .   H-  Krc  +  L  =  0     .     .     .     (1), 
in  which   the  coefficients  are  supposed  to  be  real,  then  will 
rt  —  Z>  V—  1   be  a  root  of  that  equation. 

Since  a  -{•  b  ^/  —  I   is  a  root  of  (1), 

{a  JrbV-'lT+  ^{a  +  b  A/-"ir'  +  B(«  4-  h  ^-1)"''+ 

+  K(« +  Z>V^1)  + L  =  0    .     .     .     (2). 

If  we  expand  those  terms  of  (2)  which  contain  a  -\- b  a/—  1, 
the  resulting  equation  will  contain  some  terms  which  are  real  and 


388  THEOEY    OF    EQUATIONS. 

6ome  wliich  are  imaginary.     Since  the  coefficients  A,  B,  C,  .  .  .  , 

and  the  even  powers  of  ^V— 1  are  real,  it  follows  that  V— 1 
will  occur  only  in  connection  with  the  odd  powers  of  h.  Denoting 
the  snm  of  the  real  terms  by  P,  and  the  sum  of  the  imaginary 
terms  by  Q  V—  !>  we  have 

P  +  QV^=1  =  0    .    .    .    (3); 

whence,  P  =  —  Q  \/^^    .    .    .     (4). 

To  satisfy  (4)  we  must  have  P  =  0  and  Q  =  0,  for  a  real 
quantity  cannot  be  equal  to  an  imaginary  quantity. 

Now  if  a  —  bv^—1  be  substituted  for  x  in  (1),  its  first  mem- 
ber, when  expanded,  will  differ  from  the  result  obtained  by  ex- 
panding the  first  member  of  (2)  only  in  the  sign  of  the  odd  powers 
of  b  V—  1 ;  that  is,  the  first  member  of  (1)  may  be  represented 

by  P  — QV— 1  when  a  — JV— 1  is  substituted  for  a:.  But 
P  =  0  and  Q  =  0; 

P-QV~1  =  0    .    .    .    (5). 

Therefore  a  —  bV—  I   is  a  root  of  (1). 

CoE.  1. — An  equation  of  an  odd  degree  whose  coeflBcients  are 
real  has  at  least  one  real  root. 

CoE.  2. — The  product  of  the  two  roots  a  -{-  b  V~—  1  and 
a  —  b  V^l  is  a^  +  W,  which  is  a  real  positive  quantity ;  hence, 
an  equation  of  an  even  degree  whose  coefficients  are  real,  and 
whose  absolute  term  is  negative,  must  have  at  least  two  real 
roots. 

CoE.  3.— The  product  o{ x—{a-\-bV^^^  and  x—{a—bV-i) 
is  (x  —  ay  +  Z^,  which  is  a  rational  quadratic  expression,  and 
positive  for  all  real  yalues  of  x. 

Cor.  4. — If  a  +  v^,  in  which  V^  is  a  simple  quadratic 
surd,  is  a  root  of  an  equation  whose  coefficients  are  rational,  then 
will  a  —  Vb  be  a  root  of  that  equation. 


TEANSFOEMATION    OF    EQUATIONS.  389 

EXAMPLES, 

1.  1  _  2  V^^i  is  a  root  of  the  equation  a:^— :r'^  +  3:c  +  5=0; 
what  are  the  other  roots?  Ans.  —  1  and  1  +  2  ^/  —  1. 

2.  V—  1  is  a  root  of  the  equation  a;*  +  4a:3H-ea;'^  +  4a;  +  5  =  0; 
wiiat  are  the  other  roots  ? 

3.  3  +  a/—  2  is  a  root  of  the  equation  x^-{-a^—2bx^-\-A:lx-{-QQ 
=0;  what  are  the  other  roots? 

4.  V^    is  a  root  of  the  equation    a;^  +  2a;3— 4ic2_4a;  +  4=0 ; 
what  are  the  other  roots  ? 

5.  2  +  V3  is  a  root  of  the  equation  a:*— 2a;3— 5.^2_6:c  +  2=0 ; 
what  are  the  other  roots  ? 

6.  a/3  and  1  —  2  a/— 1   are  roots  of  the  equation  a^—a^-\- 
8a;2— 9x— 15=0;  what  are  the  other  roots? 

7.  Has  the  equation  a;^— 2a;  4-4=0  a  real  root?    Why? 

8.  Has  the   equation    a."*— 4a;2-f  4.c— 1=0    any  real   roots? 
Why? 

TRANSFORMATION  OF  EQUATIONS. 

596.  To  transform  an  equation  into  another,  the 
roots  of  which  shall  be  those  of  the  proposed  equar 
tion  with  contrary  signs. 

Let  r  represent  a  root  of  the  equation 

a;"  +  Aa;"-i  +  B.^"-2  +  Qx""-^  +....=  0    .    .    .     (1) ; 

then 

r»  +  Ar«-i  +  Br«-2  +  Cr"-^  -f  ....==  0     .     .    .    (2). 

Changing  the  signs  of  (2), 

__  ,.n  _  Ar"-i  —  Br"-2  —  Cr"-^  .  .  .  .  =  0    .     .     .     (3). 

Changing  the  signs  of  the  alternate  terms  of  (1), 

^«  _  A^«-l  _|_  I3a,n-2  _  Q^n-Z  _|_    _  _    ,33  0     .     .      .      (4). 


390  THEORY     OF    EQUATIONS. 

Substituting  —  r  for  Xy  the  first  member  of  (4)  becomes 

/•«  4-  Ar"-i  +  Br" -2  _|_  Cr"-3  +...., 
or  —  r«  —  Ar"-i  —  B?'«-2  —  C?'»-3  _  .  .  .  . 

according  as  n  is  ere;i  or  orftZ.     But,  by  (2)  and  (3),  each  of  these 
expressions  is  equal  to  zero ;  hence  —  r  is  a  root  of  (4). 

Since  —  r  is  a  root  of  (4),  it  is  a  root  of  the  equation  obtained 
by  changing  all  the  signs  of  (4) ;  that  is,  —  r  is  a  root  of  the 
equation 

—  a;"  +  Aa;"-i  —  Ba;«-2  +  Ca;~-3  —....=  0    .    .    .     (5). 

Hence, 

If  the  signs  of  the  alternate  terms  of  a  complete  equation  be 
changed,  the  signs  of  all  the  roots  will  he  changed. 

An  incomplete  equation  may  be  rendered  complete  by  insert- 
ing the  missing  temis,  with  zero  for  the  coefficient  of  each  of 
them.  Thus,  by  inserting  Ooc/^  and  Oai^,  the  equation  a^  -\-Zx^  — 
4^3  ^  4a;  +  7  =  0   becomes    3^  +  ^3^-\-0x^—^-\-0x^—^x-\-^=Q' 

EXAMPLES, 

1.  The  roots  of  the  equation  a^  —  Ix^  -\-  I'dx  —  3  =  0  are 
3,  2  +  ^/3,  and  2  —  V3;  find  the  equation  whose  roots  are 
_  3,    _  2  —  V3,   and    —  2  +  V3. 

Ans.  a^  +  7a;2  4-  13a;  +  3  =  0. 

2.  The  roots  of  the  equation    a;^  _  3^  ^  3^2  _^  17^  _  18  ==  0 

are   1,    —  2,   2  +  V—  5,    and    2  —  a/—  5 ;     what  are  the  roots 
of  the  equation   x^  -\-  ^3^  -\-  Zx^  —  llx  —  1%  =  {)'i 

Ans.  —  1,   2,    —  2  —  V~-^,    —  2  +  V^^. 

3.  The  roots  of  the  equation  a;*  +  4a;3  —  a;^  —  16a;  —  12  =  0 
are  2,  —  1,  —  2,  and  —  3 ;  what  are  the  roots  of  the  equation 
—  a;*  +  4a;3  ^  ^2  _  16a;  +  12  =  0  ?  Ans.  —  2,   1,   2,   3. 

4.  The  roots  of  the  equation  a;^  —  1  =  0  are  1,  i(— 1  -f  V—  3), 
and    ■|-(— 1  — V— 3);    what  are   the  roots   of    the   equation 

Ans,  -1,   -^(-1  + V^,    -iC-l-V^^^^). 


TKANSFOEMATION     OF     EQUATIOlirS.  391 

597.  To  transform  an  equation  containing  frac- 
tional coefficients  into  another  in  -which  the  coefficients 
shall  be  integers,  that  of  the  first  term  being  unity. 

Let  the  proposed  equation  be 

a;«  4-  A^«-i  +  B:c«-2  + +  Kx-{-h:-0    .    .    .     (1), 

in  wliicli  some  or  all  of  the  coefficxnts  A,  B,  C,  .  .  .  .   are  sup- 
posed to  be  fractional. 

Assume  y  —  kx,  or  :?;  =  "-.    Substituting  |  for  x  in  (1)  and 

multiplying  the  resulting  equation  by  ^", 

?/«  4-  A/j?/"-!  +  B7cY~^  + +  K^-"-i?/  +  W  =  0   ,    .    .  (2). 

Now,  since  7c  is  arbitrary,  we  may  give  it  such  a  value  as  will 
make  the  coefficients  Ak,  Bk\  ....  Kk''-\  Lk"'  integers. 

EXAiUrLES. 

Transform  each  of  the  following  equations  into  another  in  whicli 
the  coefficients  shall  be  entire,  that  of  the  first  term  being  unity: 

1.  ^  +  |..  +  |,.  +  ^,.Hu.|  =  o    .    .    .     (1). 

Substituting  j   for  x  in  (1)  and  multiplying  ^.he  resulting 

equation  by  h\ 

ah   ,      ck^   -      ck^         qlc^      ^         .         ,^. 

y'  +  Tf  +  -ay+-fy+-h=^  ■  ■  ■  (^)- 

Assuming  h  =  bdfh,  (2)  becomes 
yi  +  adfhif  +  cWphhf-^eW?ph^y+gVd^fh^=0    .    .    .    (3). 

2.  ^+"^V-  +  -  =  0    .    .    .    (1). 

pm       m      p  ^ 

Substituting  j  for  x  in  (1)  and  multiplying  the  resulting 
equation  by  k^, 


y.  +  ^y^  +  ^^i  +  ?E  =  o  .  .  .  (2). 

^        pm         m         p  ^  ^ 


392  THEORY    OF    EQUATIONS. 

Assuming  Jc=pn,  the  L.  C.  M.  of  the  denommators,  (2) 
becomes 

ys  +  o?/2  4-  Ip^my  +  cj^^  ~Q    .    .    ,     (3). 
3.    ^_§^  +  ^.,__J_.__H_  =  o    .    .    .     (1). 

Substituting  y^  for  a;  in  (1)  and  multiplying  the  resulting 
equation  by  k^, 

^--6-2^+12^-150^-9000  =  ^     •     •     •     (^)- 

Kesolving  the  denominators  in  (2)  into  prime  factors,  we  have 

6=2x3,       12=22x3,        150=2x3x52,       9000=23x32x53. 

Assuming  ^'  =  2  x  3  x  5,   (2)  becomes 

5-2'3-5   3      5-22-32-52   ^      7-2«'38-53         13-2^-3^-5^ 
^  2-3      ^  "^        22-3       ^  2-3-52    ^  23-32-5~3 

=  0     .     .    .     (3). 

Canceling  common  factors  in  (3), 

j^_5-5z^-5-3-52y2- 7-22-32-5y-13-2-32-5  =0; 

that  is,    7/4— 25y3^375y2_i260^— 1170=0    ...     (4). 

If  we  had  assumed  k  =  9000,  which  is  the  L.  C.  M.  of  the 
denominators  of  the  given  equation,  the  coefficients  in  the  trans- 
formed equation  would  have  been  much  larger  than  those  in  (4). 

^    ^-35-^2450"^       68G0O~ 

Ans.  if  —  6?/2  +  26y  —  85  =  0. 

K       n      13^  .   21    ,       32    2       43  1         . 

^'    ^-12^  +  40^-225^'-600^-800  =  ^- 


^725.  2/5— 65y*  +  1890y3_30720/-928800^— 972000=0. 

=  0. 

Ans,  if  —  14^2  ^  \\y  _  75  _  0. 


'-  ^-V^l^-%  =  '- 


TRANSFOEMATIOK    OF    EQUATIOITS. 


393 


598.  To  transform  an  equation  into  another,  the 
roots  of  which  shall  differ  from  those  of  the  given 
equation  by  a  given  quantity. 

Let  the  proposed  equation  be 

.T"  +  A:r"-i  4-  Bx""-^  + 4-Ka;  +  L  =  0    .     .    .     (1). 

Substituting  y  -\-  h  for  x  in  (1),  we  have 

(y4_7,)n  +  A(?/  +  /i)"-i  +  B(2/H-/i)"-2  +  ...  +  K(?/  +  70  +  L=:O...(2). 

Expanding  and  reducing,  (2)  becomes 


y^  +  nh 


r~'  + 


2 


yn-^^ 


+  h'' 


+  (^^  —  1)  A^  +  Ah""-^ 

-f  B  +  B/i«-2  I  =  0  ...  (3) 

+  ... 

+  L 

The  roots  of  (3)  differ  from  those  of  (1)  by  h,  forx  =  y  +  h. 

Denoting  the  coefficient  of  r^-^  by  A',  that  of  ^""^  by  B', .... , 
and  the  independent  term  by  L',  (3)  becomes 

^n^Ay-i  +  By'-2  +  ....+jy +  K'2^  +  L'  =  o  .  .  .  (4). 

We  now  propose  to  show  that  (4)  may  be  deduced  from  (1)  by 
Synthetic  Divismi. 

Restoring  the  vahie  of  ?/,  (4)  becomes 

+  L'  =  0    .    .    .     (5). 

Now  the  first  member  of  (5)  is  identical  with  the  first  member 
of  (I);  for,  in  deducing  (5)  from  (4)  we  merely  retraced  the  steps 
by  which  (4)  was  deduced  from  (1).    Hence  the  equation 

a;«H-  Aa:«-i+  B.c«-2-f-....+J.'c2  4.Ka;  +  L==  (a:— 70~  +  A'(a:—7i)«-i 

+  B\x-UY-^-\-  . , .  ,  +;5'(x-hf-\-^{x-U)  +  U  •  •  •   (6) 
is  an  identity. 


394  THEORY    OF    EQUATIONS. 

Dividing  the  second  member  of  (6)  by  a;  —  //,  we  obtain  the 
remainder  L';  dividing  the  quotient  by  x  —h,  we  obtain  the  re- 
mainder K';  dividing  the  second  quotient  by  a:  —  //,  we  obtain 
the  remainder  J' ;  and  so  on ;  hence  if  we  treat  the  first  member 
of  (G)  or  the  first  member  of  (1)  in  the  same  way,  vv^e  shall  obtain 
the  same  remainders.  But  these  successive  remainders  are  the 
coefiBcients  of  (4).  Hence  the  coefficients  of  (4)  may  be  obtained 
from  (1)  by  the  following 

R  ULE. 

Divide  the  first  member  of  (1)  by  x  —  h,  continuing  the  oper- 
ation until  a  remainder  is  obtained  luhich  is  independent  of  x ; 
then  divide  the  quotient  by  the  same  divisor,  and  so  on,  until  n 
divisions  have  been  performed :  the  successive  remainders  will  be 
the  coefficients  of  (4). 

EXAMPLES. 

1.  Find  an  equation  whose  roots  are  less  by  2  than  those  of 
the  equation  a:*  —  4^:3  —  8a;  +  32  =  0. 

Substituting  y  +  2  for  a;  in  this  equation,  we  obtain  y^  + 
4^  —  24?/  =  0,  which  is  the  equation  required.  The  same  result 
may  be  obtained  by  Synthetic  Divisiouy  as  follows : 


l_4+0— 

8  +  32|2 

+  2-4- 

8  —  32 

—  2-4- 

16+0  1st 

rem. 

+  2  +  0- 

8 

+  0-4- 

24  2d  rem. 

+  2  +  4 

+  2  +  0  3d  rem. 

+  2 

+  4  4th  rem. 

Hence  the  required  equation  is  ?/^+  4cy^-\-  Oy^—  24y+  0  =  0. 

2.  Find  an  equation  whose  roots  are  greater  by  3  than  those 
of  the  equation   x*  +  IGa^  +  99x^  +  228a:  +  144  =  0, 


TRANSFOBMATIOISr    OF    EQUATIONS.  395 

Substituting  ?/  —  3  for  x  in  this  equation,  we  obtain  y^  + 
14^3  _^  Qy2  _  42^  =  0.  The  same  result  may  be  obtained  by  Syn- 
thetic Division,  as  follows : 

1  +  10  +  90  +  228  +  144  I  -  3 
_    3  — 39  — 180  — 144 1 
+  13  +  CO  +    48  +      0  1st  rem. 
_    3  —  30—    90 
4-  10  +  30  —    42  2d  rem. 
—    3-21 
+    7  H-    9  3d  rem. 
— _3 
-f-    4  4th  rem. 

Hence  the  required  equation  is  y^  +  42/^  +  9y^  —  42?/  =  0. 

3.  Find  an  equation  whose  roots  are  greater  by  2  than  those 
of  the  equation  x^  +  4cX^—24:X  =  0.    Ans.  y*—iy^^Sy-\-32  =  0. 

4.  Find  an  equation  whose  roots  are  less  by  3  than  those  of 
the  equation   a;*  —  12:c3  +  lUx^  _  9a:  +  7  ==  0. 

Ajis.  y^  -  37?/2  _  123?/  —  110  =  0. 

599.  To  transform  an  equation  into  another  in 
which  the  second  or  third  term  shall  not  appear. 

Since  h  in  equation  (3)  of  Art.  598  is  arbitrary,  we  may  give 
to  it  such  a  value  as  will  cause  the  second  term  of  that  equation 
to  vanish. 

A 

Assume    nh-{-A  =  0;    then    h= .     Substituting  this 

value  for  h  in  (3),  we  obtain  an  equation  of  the  form  of 
yn  ^  B'2/"-2  -f-  Cy-3  +  .  .  .  .    +  K>y  +  L'  =  0. 

If  we  assume   ''(''-'^)^''  -\- (n  -  1)  Ah  +  B  =  0,  the  third 

term  of  (3)  will  disappear. 

Cor.— The  value  of  h  which  makes  the  second  term  disappear 
may  cause  the  disappearance  of  the  third  or  some  other  term. 


396  THEORY    OF    EQUATIONS. 

In  order  that  the  third  term  may  disappear  at  tho  same  time 
with  the  second,  it  is  necessary  that  the  value  of  h  wliich  satis- 
fies the   equation    nh  +  A  =  0   shall   also  satisfy  the  equation 

n  (n  -  1)  h^  _^  (^  _  1)  A7i  +  B  z=  0.      Substituting    -  -   for  h 

.    X1-.           X-             1          w(/i  — 1)     A2       ,         ^,A2      ^       ^ 
in  this  equation,  we  have  ,, •  -^—  (w  —  1) t-B  =  0; 

whence  A^  = -.    This  equation  expresses  the  relation  which 

must  subsist  between  the  coefficients  A  and  B  in  order  that  the 
third  term  may  disappear  with  the  second. 

EXAMPLES. 

1.  Transform  the  equation  x^  —  Q>x^  +  Sx  —  2  =i  0  into  an- 
other wanting  the  second  term.  Ans.  y^  —  4ty  —  2  =  0. 

2.  Transform  the  equation  cc*  —  12a;3  +  IW  —  9a:  +  7  =  0 
into  another  wanting  the  second  term. 

Ans.  y^  —  311  y^  —  123y  —  110  =  0. 

3.  Transform  the  equation  a:^  —  Gic^  -|-  13a;  —  12  =  0  into  an- 
other wanting  the  second  term.  Aiis.  y^  +  y  —  2  =  0. 

4.  Transform  the  equation  3^-\-bx^-\-^x  — 1=0  into  two 
others,  each  wanting  the  third  term. 

139 
Ans.  ?/3  _  2^2  _  5  _  0  and  y^  +  y^ ^r^  =  0. 

5.  Can  the  equation  x^  +  Ga:^  -f  12a;  —  oG  =  0  be  transformed 
into  another  wanting  the  second  and  third  terms  ? 


THEOREM  OP  DESCARTES. 

600.  In  any  series  of  quantities  a  pair  of  consecutive  like 
signs  is  called  a  JPertnaneuce  of  signs,  and  a  pair  of  consecu- 
tive unhke  signs  is  called  a  Variation  of  signs.  Thus,  in  the 
expression  x^  —  S-c^  _  4a:«  +  7a;5  _j.  3^^  2:x^—  x^—  a;  +  1,  there 
are  four  permanences  and  four  variations. 

601.  If  the  equation  f{x)=0  is  complete,  the  sum  of  the  num- 
ber of  permanences  and  the  number  of  variations  in  the  signs  of  the 
terms  of /(a;)  is  equal  to  the  greatest  exponent  of  a;  in  the  equation. 


THEOREM    OF    DESCAETES.  397 

602.  TJieorem  of  Descartes, — 77ie  mimber  of  real  pos- 
itive roots  of  the  equation  f(x)  =  0  cannot  exceed  the  numher  of 
variations  in  the  signs  of  its  terms  ;  and,  if  the  equation  f{x)  —  {) 
is  complete,  the  number  of  real  negative  roots  cannot  exceed  the 
numher  of  permanences  in  the  signs  of  its  terms. 

Eepresent  the  real  positive  roots  of  the  equation 
f{x)=Q    ...    (1) 

by  «,  5,  c  .  .  .  . ,  and  suppose  (1)  to  be  divided  by  the  product  of 
all  the  factors  x  —  a,  x  —  h,  x  —  c,  .  .  .  .  corresponding  to  the 
real  positive  roots  (586).     Eepresent  the  resulting  equation  by 

/,(i)  =  0    .    .    .     (2). 

This  equation  has  no  real  positive  roots. 

We  shall  now  show  that  if  (2)  be  multiplied  by  the  factor 
X  —  a  corresponding  to  a  real  positive  root,  the  number  of  varia- 
tions of  the  resulting  equation  will  be  at  least  one  greater  than 
in  (2). 

I.  Suppose  (2)  to  be  complete,  and  let  the  signs  of  its  terms  be 


The  signs  of  the  multiplier  are 


The  signs  of  the  product  are        +  ±  —  T  T  H . 

A  double  sign  is  placed  where  the  sign  of  any  term  in  the 
product  is  ambiguous. 

Now,  taking  the  ambiguous  signs  as  we  please,  the  number  of 
variations  in  the  product  is  greater  than  in  the  multiplicand; 
and  this  is  still  true  if  we  suppose  some  or  all  of  the  terms  having 
ambiguous  signs  to  vanish. 

11.  If  (2)  is  incomplete,  reduce  it  to  a  complete  form  by  in- 
serting the  missing  terms  with  zero  for  the  coefficient  of  each ; 
the  resulting  equation  will  contain  at  least  as  many  variations  as 
(2).  Multiplying  the  completed  equation  by  x  —  a,  the  number 
of  variations  in  the  product  will  be  greater  than  in  the  multipli- 
cand (I).     But  the  product  thus  obtained  is  the  same  as  the  pro- 


+  +  - 
+  -. 

+• 

+  +  - 

—  +. 
+  +  +  -. 

398  THEORY    OF    EQUATIONS. 

duct  of  /j (x)  and  x  —  a;  hence,  the  number  of  variations  in  the 
product  of  f^{x)  and  a;  —  a  is  greater  than  in  fi{x). 

We  have  thus  shown  that  when  the  factor  x  —  a  is  introduced 
into  (2),  the  resulting  equation  contains  at  least  one  more  varia- 
tion than  (2).  In  like  manner  it  may  be  shown  that  when  the 
factor  a;  —  ^  is  introduced  into  the  resulting  equation,  at  least  one 
more  variation  is  introduced ;  and  so  on. 

Hence  the  number  of  real  positive  roots  of  the  equation 
f(x)  =  0  cannot  exceed  the  number  of  variations  in  the  signs  of 
its  terms. 

We  prove  the  second  part  of  the  theorem  as  follows : 

Suppose  (1)  to  be  complete,  and  let  the  signs  of  its  alternate 
terms  be  changed ;  then  the  signs  of  the  roots  will  be  changed 
(596),  the  permanences  will  become  variations,  and  the  varia- 
tions will  become  permanences.  But  the  number  of  real  positive 
roots  of  the  resulting  equation  cannot  exceed  the  number  of  vari- 
ations in  the  signs  of  its  terms ;  hence  the  number  of  real  negative 
roots  of  the  given  equation  cannot  exceed  the  number  of  perma- 
nences in  the  signs  of  its  terms. 

Cob.  1. — Whether  the  equation  f{x)  =0  is  complete  or  not, 
its  roots  are  numerically  equal  to  those  of  the  equation/(— a:)  =0 ; 
but  the  signs  of  the  two  sets  of  roots  are  opposite.  Hence  the 
number  of  real  negative  roots  of  the  equation  /{x)  =  0  is  equal 
to  the  number  of  real  positive  roots  of  the  equation  f  (—  x)  =  0. 
But  the  number  of  real  positive  roots  of  the  equation  /(—  x)^0 
cannot  exceed  the  number  of  variations  in  the  signs  of  its  terms. 
We  may  therefore  state  the  theorem  of  Descartes  as  follows  • 

The  number  of  real  positive  roots  of  the  equation  f(x)=zO 
camiot  exceed  the  nuniber  of  variations  in  the  signs  of  f  {x),  and 
the  number  of  its  real  negative  roots  cannot  exceed  the  number  of 
variations  in  the  signs  of  f{—  x). 

Elustratio7i.—The  equation  a^  -\-  3a^  -\-  5x  —  7  =  0  has  only 
one  variation  of  signs ;  therefore  it  cannot  have  more  than  one 
real  positive  root.  By  putting  —  x  in  the  place  of  x,  we  obtain 
the  equation  x^  -{- 3x^  —  5x  —  If  =  0.  This  equation  has  only 
one  variation  of  signs;  therefore  it  cannot  have  more  than  one 


THEOREM    OF    DESCAETES.  399 

real  positive  root ;  lience  the  original  equation  cannot  have  more 
than  one  real  negative  root. 

Cor.  2.— If  the  equation  f{x)  =  0  is  complete,  and  all  its 
roots  are  real,  the  number  of  positive  roots  is  equal  to  the  number 
of  variations  in  the  signs  of  its  terms,  and  the  number  of  negative 
roots  is  equal  to  the  number  of  permanences  in  the  signs  of  its 
terms. 

Denoting  the  number  of  permanences  by  p,  the  number  of  va- 
riations by  V,  the  number  of  positive  roots  by  P,  the  number  of 
negative  roots  by  N,  and  the  highest  exponent  of  x  in  f{x)  by  n, 
we  have  v  -\-  p  =  n  and  P  +  N  =  ?i ;  hence  v  +  j9  =  P  +  N. 
Now  P  cannot  exceed  v,  and  N  cannot  exceed  p ;  hence  P  =  v, 
and  N  ^=  p. 

Cor.  3. — By  means  of  the  theorem  of  Descartes  we  can 
sometimes  detect  the  presence  of  imaginary  roots  in  an  equa- 
tion. 

Illustration. — The  equation  a;^  +  16  =  0  has  no  variation  of 
signs ;  therefore  it  has  no  real  positive  root.  By  putting  —  a;  in 
the  place  of  x,  we  obtain  the  equation  x^  -f  16  =  0.  This  equa- 
tion has  no  variation  of  signs ;  therefore  it  has  no  real  positive 
root;  hence  the  original  equation  has  no  real  negative  root. 
Therefore  the  roots  of  the  equation  a;^  -f  16  =  0  are  imaginary. 


EXAMPLES. 

1.  Show  that  the  equation  x^  -{-  5x  -{-  IS  =  0  has  only  one 
real  root. 

2.  All  the  roots  of  the  equation   x^  -\-  5x^  -{-  2x  —  S  =  0  are 
real;  how  many  of  them  are  negative?  Ans.  Two. 

3.  All  the  roots  of  the  equation  x^  —  ^x'^—llx^  +  29x  +30=0 
are  real ;  how  many  of  them  are  positive  ?  Ans.  Two. 

4.  All  the  roots  of  the  equation   x^— 3x^^—50^ -{-16x^-}-^x— 12 
=  0   are  real ;  how  many  of  them  are  positive  ?       Ans.  Three. 


400 


THEORY    OF    EQUATIONS. 


DERIVED    FUNCTIONS. 


603.  Substituting  x  -{-  h  for  x  in  the  identity 
f{x)  =  a;»  +  A2;"-i  +  Ba:'»-2  + +  Kic  +  L    . 


(1), 


and  arranging  the  result  according  to  the  ascending  powers  of  /^, 
we  obtain 


f{x  +  h) 


=        x" 


n3f*~^ 


+  K 


h  H-  n{n—l)x**~^ 

+  (n— l)(?i— 2)Aa:«-3 


(2). 


7^2 


Denoting  the  coefficient  of  h  by  /'(:z^),  that  of  --—  by  f"{x)j 
and  so  on,  (2)  may  be  written 


f{x  +  h)^f{x)+f(x)h+r{x)'-^+f'\x)  ^+  .  . . . 


(3). 


The  expression  f{x)  is  called  the  primitive  fimction,  the 
expressiony'(.r)  is  called  the  Jirst  derived  function,  or  simply  the 
first  derivative,  i\\Q,  expression /'(a:)  is  called  the  second  deriva- 
tive, and  so  on. 

The  first  derivative  may  be  obtained  from  the  primitive  func- 
tion by  multipl}ing  each  of  its  terms  by  the  exponent  of  x  in  that 
term  and  dividing  the  result  by  a;;  the  second  derivative  may  be 
obtained  from  the  first  in  the  same  way  that  the  first  is  obtained 
from  the  primitive  function ;  and  so  on. 


EXAMPLES. 


1.  Find  the  derivatives  of  a;^  _  q^z  ^  Sa;  —  2. 

(  1st.  Sx^  —  Ux-}-  8, 
Ans.  I  2d.    (jx  —  12, 
(  3d.    6. 


DERIVED    FUNCTIONS.  4:01 

2.  Transfoiin  the  equation  a^  —  Gx^  -\-  Sx  —  2  =zO  into 
another  wanting  the  second  term. 

Substituting  x-\-2  for  x  in  the  identity/ (2-)  =:a;2 — G.^2_|-8:c— 2, 

we  have 

22         23 
f{x-^2)  =  x^-6a^-\-Sx—2-^{3xi—12x  +  8y2-{-(6x—12)^-\-6~ 

—  a^  —  4:X  —  2;    hence  the  required  equation  is    a^ — ix — 2  =  0. 

3.  Find  an  equation  whose  roots  are  less  by  1  than  those  of  the 
equation   x^  —  2^:2  _|.  3^  —  4  =  0. 

Substituting  x-^1  for  x  in  the  identity/(a:)=a:3 — 20^24- 3a; — 4, 
we  have 

/(rr +  1)  =a;3_22;2-f3a;—4  +  (3a;2—4a;  +  3)l  +  (6a;— 4)^  +  6^ 

=  x^  -{-  x^  -\-2x  —  2;  hence  the  required  equation  is 

xi-^x^-\-2x  —  2  =  0. 

Let  the  student  solve  all  the  examples  of  Art.  599  by  the 
method  of  derived  functions. 

604.  TJie  first  derivative  of  the  product  of  tivo  ftmctions  of 
the  same  quantity  is  equal  to  the  sum  of  the  products  obtained  hy 
multiplying  each  hy  the  first  derivative  of  the  other. 

Substituting  x  -\- h  for  x  in  the  two  expressions  f{x)  and 
/i  (^)>  w®  obtain 

f(x-\-h)=  f{a)-\-f{x)h+ (1)     (603), 

and       fr{^+h)=f^{x)-{-f^{x)h-\- (2). 

Multiplying  (1)  by  (2), 
f{x  +  h)f^{x+n)  =f{x)f,{x)+f,(x)f\x)h+f{x)f^{x)h  +  .... 

.    .     .     (3). 

The  coefficient  of  h  in  (3)  is  the  sum  of  the  products  obtained 
.by  multiplying  /^  {x)  by  the  first  derivative  of  /  {x)  and  /  {x)  by 
the  first  derivative  0^  f^(x)  and  this  coefficient  is  the  first  deriva- 
tive of /(2;)/i  (2:)    (603). 
26 


402  THEORY    OF    EQUATIONS. 

Cor. — In  like  manner  it  may  be  shown  that  the  first  deriva- 
tive of  the  product  of  three  or  more  functions  of  the  same  quan- 
tity is  equal  to  the  sum  of  the  products  obtained  by  multiplying 
the  first  derivative  of  each  by  the  product  of  the  other  functions. 

EXAMrZES. 

Find  the  first  derivative  of  each  of  the  following  expressions : 

1.  x^  (x  —  ay.  Ans.  2x  (x  —  af -\-  3{x  —  a)- x\ 

2.  (a  -^  x){l)  +  x).       Ans.  J)-^x+a-\-x  =  a-\-h  +  'ilx, 

3.  {x-aY  (x-by.  Ans.  2{x-a){x-bY-i-.S(x—by  {x-ay. 

4.  {x  —  ay{x  —  by(x  —  cy. 

Ans.  3(x—ay  (x^bf  (x—cy+4:(x—bf  {x—ay  {x—cf 
+  b{x—cy(x—ay  {x—by. 

5.  {x  —  ay  {x  —  b)"". 

Ans,  n  (x—ay~^  {x—b)^  +  m(x-'by"~^  {x—ay. 

ROOTS    COMMON    TO    TWO    EQUATIONS. 

605.  If  a  is  a  root  of  the  equation  f{x)  =  0,  f{x)  is  divisible 
by  X  —  a,  and  if  a  is  a  root  of  the  equation  f^ix)  =  0,  fi{x)  is 
divisible  by  x  —  a;  hence  the  roots  of  the  equation  obtained  by 
putting  the  G.  C.  D.  of  f(x)  and  f^ix)  equal  to  zero  will  be  the 
roots  common  to  the  two  equations  f{x)  =  0  and  fi{x)  =  0. 

EXAMPLJES. 

1.  Find  the  root  which  is  common  to  the  two  equations 

X^—2x^—7x^  +  20x—V2=0    and     A3^—6x^—Ux  +  20=0. 

The  G.  C.  D.  of  the  first  members  of  these  equations  is  x—2; 
hence  2  is  a  root  common  to  the  given  equations. 

2.  Find  the  roots  common  to  the  two  equations 

a;4— 22:3— llic2  4-122;  +  36=0    and     ^a^—6x^—22x-\-12=0. 

Ans.  3  and  —2. 


EQUAL    ROOTS.  403 

3.  Find  the  roots  common  to  the  two  equations 
x'^—Sx^-\-c(P—4^-  +  12x—4:=zO    and     2x^—Qx^  +  3x^—3x-^l=0. 

Ans.  ^r . 

4.  How  many  roots  are  common  to  the  two  equations 
a^—4,Q:^^e^x^-\-10a^—26x—4:=0  and  2x^—18x^-\-d9x^—2bx^ 
+  a;  +  1  =  0  ?  Ans.  Three. 

EQUAL    ROOTS. 

606.  The  equation  f{x)=0  is  called  the  Primitive  Equa- 
tion, and  the  equation  f'{z)  =  0,  which  is  obtained  by  putting 
the  first  derivative  of  f{x)  equal  to  zero,  is  called  the  First 
Derived  Equation, 

607.  If  a  root  occurs  n  times  in  the  equation  f{x)  =  0,  it 
will  occur  n  —  1   times  i7i  the  equation  f(x)  =  0. 

Let  the  proposed  equation  be 

f(x)  =  {x-ay{x-b)(x-c),,,.=0    .    .    .    (1), 
in  which  a  occurs  as  a  root  n  times. 

The  first  derivative  of  each  of  the  factors  x  —  b,  x  —  c, . , , , 
is  1 ;  hence  the  first  derived  equation  is 

f'{x)  =n{x  —  «)"~^  {x  —  i)  (x  —  c)  .  .  .  .  -\-  {x  —  aY{x  —  c) . . . . 

^  {x-aY{x-b)  ....  -\-  =  Q    .    .    .     (2), 
in  which  a  occurs  as  a  root  n  —  1  times  (587). 

Cor. — A  root  which  occurs  only  once  in  (1)  does  not  occur  in 
(2) ;  hence  any  root  which  is  common  to  (1)  and  (2)  is  one  of  the 
equal  roots  of  (1). 

Find  all  the  roots  of  each  of  the  following  equations : 

1.    f(x)—x^—W-\-Ux  —  VZ  =  0. 

f\x)  =  3xi  —  Ux  +  16  =  0.  The  G.  C.  D.  of  f(x)  and 
f'{x)  is  X  —  2.  Putting  this  equal  to  zero,  we  have  x  —  2  nr  0 ; 
whence  x  =  2.  The  given  equation,  therefore,  has  two  roots 
equal  to  2.  The  remaining  root  of  the  given  equation  may  be 
found  by  the  principle  of  Art.  588^  Cor. 


404  THEORY    OF    EQUATIOKS. 

2.  x^-  ll3^  +  Ux^  -  76a;  +  48  =  0.  Ans,  2,  2,  3,  4. 

3.  2x^^123^ -\-ldxi—Gx+ 9=0.  Ans.  3,  3,  ±  \/ -  )- 

2 

4.  7^  —  2x^  +  ^3?— Ix^  -\-^x—^  =  0. 

Ans.  1,  1,  1,  -i±|A/^^ni. 

6.    /(a;)=a;7— 9a:5^C2:4  +  15a;8— 12a?2-7a;+6=0    .    .    .     (1). 
The  first  derived  equation  is 
f'{x)  =  W-^  45a:*  +  24a;8  +  45a:3  _  24a;  —  7  =  0    .     .    .     (2). 

The  G.  C.  D.  of  f{x)  and  f'{x)  is  a^-x^-x  +  \.  Equat- 
ing  this  with  zero,  we  have 

a^-x^^x-\-\=0    .    .    .     (3). 

The  G.  C.  D.  of  the  first  member  of  (3)  and  its  first  derivative 
is  a;  —  1.  Equating  this  with  zero,  we  find  :?;  =  1 ;  hence  (3)  has 
two  roots  equal  to  1.    The  remaining  root  of  (3)  is  —  1  (593). 

Now,  since  (3)  has  two  roots  equal  to  1,  and  one  root  equal  to 

—  1,  (1)  must  have  three  roots  equal  to  1  and  two  roots  equal  to 

—  1.  Dividing  f(x)  by  {x  —  1)8  (a;  +  1)^  we  obtain  x^-\-x—(j. 
Equating  this  with  zero,  we  have  7?  +  x  —  6  =  0;  whence, 
a;  =  2  or  —  3.  The  roots  of  (1)  are  therefore  1, 1,  1,  —  1,  —  1, 
2,  —3. 

6.  0^  —  2x^  —  23?  +  4.7?  ■]-x  — 2  =  0, 

7.  a;^  —  6a;*  +  4a;3  +  ^x^  —  12a;  +  4  =  0. 

LIMITS  OF  THE  ROOTS  OF  AN  EQUATION. 

608.  If  the  coefficients  of  f{x)  are  real,  and  the  results  oh- 
tained  Inj  snhstituting  p  and  qfor  x  in  f{x)  have  Wee  signs,  the 
equation  f(x)  =  0  has  either  no  root  or  an  even  number  of  roots 
lying  heticeen  jp  and  q  ;  hut  if  the  results  have  contrary  signs,  the 
equation  has  an  odd  number  of  roots  lying  between  p  and  q. 


LIMITS    or    THE    ROOTS    OF    AX    EQUATIOIsT.  405 

Let  the  real  roots  of  the  equation    f{x)  =  0.,.(l)    be  de- 
noted by    «,  by  c,  . .  .  .  Jc,    and  let  the  quotient  obtained  by  divid- 
ing /(re)   by   {x  —  a){x  —  b)(x  —  c)  ,  .  .  .  (x  —  k)  he  denoted 
by  /iW;  then 
f(x)  =  {x-a)ix-i){x-c) {x-k)Mx)     .    .     .     (2). 

Now,  since  /^  (x)  is  the  product  of  all  the  factors  correspond- 
ing to  the  imaginary  roots  of  (1),  and  since  the  number  of  these 
imaginary  roots  is  even,  it  follows  that  /^  (x)  is  positive  for  all 
real  values  of  x  (595,  Cor.  3). 

Substituting  jj  and  q,  in  succession,  for  x  in  (2),  we  have 

f(p)  =  {p-a){p-b){p-c)  .  .  .  .  (p-Ic)Mp)    .    .    .     (3), 
/(?)  =  (?-«)  (?-*)(?-<■•) (<?-*)/, (?)       .    •    •    (*)• 


Dividing  (3)  by  (4), 
liEl _  P-'^ .  EszA  .  P^zl  P-^ .  Mil 

f{q)        q  —  a'q  —  b'q—c''''q  —  k'  J\{q) 


(5). 


/(;') 


Suppose  f{p)   and  f{q)   have  like  signs ;   then   -^^A-    will 

be  positive ;  and  since  44^  is  positive,  either  all  the  factors 

p—a     p  —  h     p  —  c  p—Jc  ,   .  ...  ,, 

^ , ^, , ..... Y    must  be  positive,  or  the 

q—aq—hq—c  q—k  ^ 

number  of  negative  factors  must  be  even.     If  all  the  factors  are 

I      positive,  no  root  of  (1)  can  lie  between  p  and  q ;  for,  if  possible, 
suppose  the  root  h  lies  between  p  and  q ;  then  p  —  h  and  q  —  h 

ip J) 

would  have  contrary  signs;  therefore would  be  negative. 

If  the  number  of  negative  factors  is  even,  then  (1)  has  an  even 
number  of  its  roots  lying  between  p  and  q ;  for,  if  any  factor,  as 

7)  —  C 

,  is  negative,  then  p  —  c  and  q  —  c  must  have  contrary 

q      c 

signs ;  therefore  c  lies  between  j9  and  </. 

Suppose  f{p)   and  f{q)   have  contrary  signs;    then    4./'^ 
will  be  negative,  and  the  number  of  negative  factors  must  be  odd. 


406  THEORY    OF    EQUATIONS. 

D  ~~  C 

But  when  any  factor  as is  negative,  the  root  c  lies  be- 
tween p  and  q ;  hence  (1)  has  an  odd  number  of  its  roots  between 
p  and  q  when  f{p)    and  f(q)   have  contrary  cigns. 

Cor.  1. — If  p  is  less  than  the  least  root  of  the  equation 
f(x)  =  0,  f(p)  will  be  positive  or  negative,  according  as  the  de- 
gree of  the  equation  is  even  or  odd. 

Cor.  2. — If  q  is  greater  than  the  greatest  root  of  the  equation 
f{x)  =  0,  f(q)   will  be  positive. 

Cor.  3. — Suppose  that  (1)  has  no  equal  roots,  and  that  a  is 
the  smallest  of  its  real  roots,  b  the  next  smallest,  and  so  on.  Now 
suppose  X  to  assume,  in  succession,  every  possible  value  from  —  oo 
to  H-  oo;  then  the  sign  of  f{x)  will  change  from  +  to  — ,  or 
from  —  to  4-,  as  often  as  x  passes  a  real  root  of  the  equation; 
for  as  long  as  a;  is  less  than  a,  all  the  factors  x—afX—b,x—c,.... 
X — k  are  negative;  but  when  x  becomes  greater  than  a  and  less 
than  bf  the  factor  x  —  a  will  be  positive,  while  the  other  fiictors 
X  —  bfX  —  Cy..,.x  —  k  will  be  negative.  In  like  manner  it 
may  be  shown  that  the  sign  of  f(x)  changes  when  x  passes 
either  of  the  other  real  roots. 


EXAMPLES. 

1.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 
fix)  =a^^.xi^x-100  =  0. 

When  x  =  i,  f(x)  is  negative;  and  when  x  =  5,  f(x)  is 
positive ;  hence  there  must  be  a  root  between  4  and  5 ;  that  is,  4 
is  the  first  figure  of  one  of  the  roots. 

2.  Find  the  first  figure  of  each  of  the  roots  of  the  equation 
a;3  _  3,^.2  _  12a;  +  24  =  0.  Ans.  1,  4,  -  3. 

3.  Find  the  first  figure  of  each  of  the  roots  of  the  equation 
a4  _  i2a;2  ^_  12a;  —  3  =  0.  Ans.  2,  .6,  .4,  —  3. 

4.  Find  the  first  figure  of  each  of  the  roots  of  the  equation 
sfi  —  lOieS  ^  62:  -M  =  0.  Ans.  -  3,  —  .6,  —  .1,  .8,  3. 


LIMITS    OF    THE    ROOTS    OF    AN    EQUATION.  407 

609.  To  find  a  superior  limit  of  the  positive  roots 
of  an  equation. 

Suppose  the  equation  to  be  of  the  v}^  degree.  Denote  the 
negative  coefficient  wliose  absolute  value  is  the  greatest  by  —  P, 
the  exponent  of  x  in  the  negative  term  of  highest  degree  by  m, 
and  the  absolute  value  of  the  sum  of  all  the  negative  terms  by  N ; 

then 

N  <  P  +  Pa;  +  Pir2  + +  P.^»«    .    .     .     (1). 

But  V-\-Vx-^V^^. . .  .  +  Pa;^:=^_-|-^    .    .    .    (2)  (520);. 

p.^OT+l  p  p^ni+l 

N<:^^^ T--<^     •    •     •     (3)- 

X  —\  x  —  \  ' 

Now,  the  absolute  value  of  the  sum  of  all  the  negative  terms 
of  the  given  equation  is  equal  to  the  sum  of  all  the  positive  terms; 

hence must  be  greater  than  any  positive  term  as  x^ ;  that  is, 

X  —  1 

Py7n+1 

-''<f4n  •  •  •  (^)- 

X 1 

Multiplying  both  members  of  (4)  by  -^7]-, 

X 

(a;  — l).'r«-"^-i<P    .     .     .     (5). 
But  x  —  \<Cx\    hence, 

(2;  _  l)«-"»-i  <  a;"-^-i    .     .     .     (6). 
Multiplying  both  members  of  (6)  by  x—\, 

(a;_l)n-m<^(a;_l):j;n-m-l      .       .      .       (7) ; 
(2;_l)n-n»<p      .      .      .       (8); 

whence,  a;  —  1  <     VP, 

or  a:  <  1  +     V  P. 

Denoting  this  superior  limit  of  the  positive  roots  by  L,  we  have 


408  THEOEY    OF    EQUATIONS. 

EXAMPZE8, 

Find  a  superior  limit  of  the  positive  roots  in  each  of  the  fol- 
lowing equations : 

1.  7^  -\-  hx^  +  23^  —  14^2  —  26a;  +  10  =  0. 

In  this  equation  w  =  5,  ?7i  ==  2,  and  P  =  26 ;  hence, 

L=H-V26. 

2.  a^^hs?  —  25rJ  —  VZx  +  68  =  0.  Ans,  6. 

3.  a^  —  bx^  —  9a;  +  12  =  0.  Ans.  4. 

4.  a:3  ^  2^  _|.  3c  _  3  _  0.  Ans.  3. 

610.  To  find  an  inferior  limit  of  the  positive  roots 
of  an  equation. 

Substitute  -  for  x  in  the  given  equation,  and  find  a  superior 

limit  of  the  positive  values  of  y  in  the  resulting  equation.     Denote 
this  limit  by  L' ;  then 

whence,  -  >  t-/  > 

y     ^ 

that  is,  a;  >  th  • 

Li 

Hence   p  is  an  inferior  limit  of  the  positive  roots  of  the  given 

equation. 

EXAXPZES. 

Find  an  inferior  limit  of  the  positive  roots  in  each  of  the  fol- 
lowing equations : 

1.    a;5+ 5a;*  + 2a:3_i4a;2_26a;+ 10  =  0. 

Substituting  -  for  x  and  reducing,  we  have 

-      26    .      14   3       2    2       5  1        . 

2^-l0  2^-10^  +10  2^'  +  io^+Io  =  ^- 

A  superior  limit  of  the  positive  roots  of  this  equation  is  3.6; 
hence  ^-  is  an  inferior  limit  of  the  positive  roots  of  the  given 
equation. 


stusm's  theorem.  409 

o 

2H-V40 

3.  a:«  —  5^:5  _|.  ^4  +  12a:3  _  i^x^  4-1  =  0. 

4.  3^  —  ^3^  +  12a;2  +  IQx  —  39  =  0. 

611.  To  find  the  limits  of  the  negative  roots  of  an 
equation. 

Substitute  —  x  for  x  in  the  given  equation,  and  find  the  limits 
of  the  positive  roots  of  the  resulting  equation.  By  changing  the 
signs  of  these  Hmits  we  obtain  the  limits  of  the  negative  roots  of 
the  given  equation  (602,  Coe.  1). 

EXAMPLES. 

Find  the  limits  of  the  negative  roots  in  each  of  the  following 
equations : 

1.  3^  —  ^X^  +  bX-{-l  =  0.  ^W5.   —  (l  + V7),    —  :^. 

2.  ic*  —  15a?^  —  lOz  4- 24  =  0. 

3.  2;6__3a;5_|.2a;4  4_27a:3  — 4a;2— 1  =  0. 

STURM'S    THEOREM. 

612.  If  the  coefficients  of  /  {x)  are  real  and  the  equation 
f{x)  =  0  has  no  equal  roots,  then,  if  x  is  made  to  assume,  in  suc- 
cession, all  real  values  from  —  oo  to  +  oo,  the  sign  of/ (a;)  will 
change  as  often  as  x  passes  a  real  root  of  the  equation  (608, 
Cor.  3).  Sturm's  Theorem  enables  us  to  determine  the  number 
of  such  changes  of  sign. 

613.  Stiirtn^s  Functions.— h^i  f{x)z=zO  be  an  equa- 
tion whose  coefficients  are  real,  and  which  is  freed  from  equal 
roots  (607) ;  and  \etf{x)  be  the  first  derivative  offix). 

We  now  apply  to  f(x)  and  f'{x)  the  process  of  finding  their 
G.  C.  D.  (125),  with  this  modification,  namely:  1.  When  a 
remainder  is  found  which  is  of  a  lower  degree  than  the  correspond- 
ing dividend  and  divisor,  ice  change  its  sign  and  use  the  result 
for  the  next  divisor.  2.  We  neither  introduce  nor  reject  a  nega- 
tive factor  in  preparing  for  division. 


410  THEORY    OF    EQUATIONS. 

We  continue  the  operation  until  a  remainder  is  obtained  which 
is  independent  of  a:,  and  change  the  sign  of  that  remaiader  also. 

Let  /i(:r),  f^(x),  f^(x), f^ix)  denote  the  series  of  modi- 
fied remainders  thus  obtained. 

The  functions  f{x),  f\x),  f,{x),  f,(x),  f,{x), . .  .  .^x)  are 
called  Sturm^s  Function^. 

The  functions  f'(x),  /^(.r),  f^{x),  f^(x), f^{x)  are  called 

Auxiliary  Functions, 

614.  Stiirm^s  Theorem, — If  x  he  conceived  to  assume, 
in  succession,  all  real  values  from  — -  oo  to  -f  oo ,  there  will  be 
no  change  in  the  number  of  variations  in  the  signs  of  the  scries  of 

functions  f  {x),  f{x),  fiix),f^(x),  f^(x), f„(x)y  except  lohen 

X passes  through  a  real  root  of  the  equation  /(.r)  =  0  ;  and  when 
X  passes  through  such  a  root,  there  will  he  a  loss  of  only  one  varia- 
tion. 

I.  fn{x)  is  not  zero;  for,  by  hypothesis,  it  is  independent  of  a;; 
hence,  if  it  were  zero,  /  (x)  and  f'{x)  would  have  a  common  di- 
visor, and  the  equation. /(a:)  =  0  w^ould  have  equal  roots  (607) ; 
but  this  is  contrary  to  the  hypothesis. 

II.  Two  consecutive  functions  cannot  vanish  for  the  same 
value  of  X, 

Let  q^,  q^,  q^, .  . .  ,  qn  denote  the  successive  quotients  ob- 
tained by  performing  the  operations  described  in  Art.  613 ;  then, 
by  the  principles  of  division, 


/(x)=(7,.f(.r)-/,(x) 
/i(a^)=?3/s(^)-/3(*) 


f.-,(x)=q,f,-,{x)-f,{x) 


(2), 
(3), 


(«). 


Now  suppose  f'{x)  and  fi{x)  to  vanish  at  the  same  time; 
then  by  (2)  we  shall  have  f2(x)  =  0 ;  hence  by  (3),  f^ix)  =  0 ; 
and  so  on ;  that  is,  if  two  consecutive  functions  vanish  at  the  same 
time,  all  the  succeeding  functions^  including  f„{x)  would  vanish ; 
but  this  is  impossible  (I). 


411 

III.  When  any  auxiliary  function  vanishes,  the  two  adjacent 
functions  have  contrary  signs.  Thus,  if  f^^^)  ■=  0,  we  have  by 
(3),/.(.^)  =  -/3W- 

IV.  No  change  can  be  made  in  the  sign  of  any  one  of  Sturm's 
functions,  except  when  x  passes  through  a  vahie  which  causes 
that  function  to  vanish  (608,  Cor.  3). 

V.  Sturm's  functions  neither  gain  nor  lose  a  variation  of  signs 
when  X  passes  through  a  value  which  causes  one  or  more  of  the  aux- 
ihary  functions  to  vanish,  but  which  does  not  cause /(a;)  to  vanish. 

1.  Suppose  /i  {x)  vanishes  when  x-=.c,  and  that  no  other 
function  vanishes  for  this  value  of  x.  Let  h  be  a  positive  quantity 
so  small  that  no  one  of  Sturm's  functions  except  /^  {x)  vanishes 
while  X  is  passing  from  c  —  li  to  c  •\-  h. 

When  x  =  c,  f'{x)  and  /g  {x)  have  contrary  signs  (III) ; 
hence  they  have  contrary  signs  all  the  time  that  x  is  passing  from 
c  —  h  to  c  -\-h  (IV).  Now  at  the  instant  x  becomes  equal  to  c, 
f^(x)  changes  its  sign  (608,  Cor.  3) ;  hence,  before  the  change, 
its  sign  is  like  that  of  one  of  the  adjacent  functions,  and  after 
the  change  it  is  like  that  of  the  other.  But  no  change  in  the 
number  of  variations  of  signs  in  a  row  of  signs  can  be  made  by 
simply  changing  a  sign  situated  between  two  adjacent  contrary 

signs.     Thus,  in  the  row  of  signs    -\ 1 1 1 

there  are  seven  variations;  and  if  we  change  the  fourth  sign  there 
are  still  seven  variations. 

Hence  Sturm's  functions  neither  gain  nor  lose  a  variation  of 
signs  while  x  is  passing  from  c  —  h  to  c  -\-  h. 

2.  Suppose  that  when  /j  (x)  vanishes,  other  auxiliary  functions 
vanish.  The  vanishing  functions  cannot  be  consecutive  (II) ;  the 
functions  adjacent  to  each  vanishing  function  have  contrary  signs 
while  X  is  passing  from  c  —  h  to  c  -\-  h;  and  each  vanishing  func- 
tion changes  its  sign  at  the  instant  x  becomes  equal  to  c.  But,  as 
we  have  just  shown,  this  change  of  sign  does  not  change  the  num- 
ber of  variations  in  the  row  of  signs. 

VI.  Sturm's  functions  lose  one  variation  of  signs,  and  only 
one,  each  time  x  passes  through  a  real  root  of  the  equation 
f(x)  =  0.  Let  a  be  a  real  root  of  the  equation  f{x)z=0;  thep 
'f{a)=0. 


412  THEORY    OF    EQUATIONS. 

Substituting  a  +  h  for  x  in  f{x)  and  f'{x),  and  developing 
by  Art.  603,  we  bave 

/(a  +  /0=/^(/>)+/>)|-+/>)|+ ••'•)   •   •   •   (1)' 

7,2 

f\a  +  li)=     na)+f'\a)h^f"'{a)^^ (2). 

Now  assume  tbe  absolute  value  of  h  to  be  so  small  that  the 
first  term  in  each  of  these  developments  shall  be  numerically 
greater  than  tlie  sum  of  the  other  terms;  then  the  sign  of 
f{a  4-  h)  will  be  the  same  as  that  of  hf\ci),  and  the  sign  of 
f\a  +  1i)  will  be  the  same  as  that  of  f\a).  Hence  f{x)  and 
f'(x)  will  have  contrary  signs  when  h  is  negative,  and  like  signs 
when  h  is  positive.  But  when  h  is  negative,  x  is  less  than  a,  and 
when  /*  is  positive,  x  is  greater  than  a ;  hence  when  x  passes  a  real 
root  of  the  equation  f{x)  =  0,  a  variation  is  changed  into  a  per- 
manence. Now  it  is  evident,  from  (2),  that  f\x)  cannot  vanish 
as  long  as  h  has  such  a  value  that  f\a)  is  numerically  greater 

than  f"{a)h  +/'"(«)  uT  +  •  •  •  •     Some  of  the  auxiliary  func- 

tions  lying  between  f'{x)  and  fj^x)  may,  however,  vanish  and 
change  signs  while  x  is  passing  through  the  root  a ;  but  the  change 
of  a  sign  lying  between  two  adjacent  contrary  signs  (III)  does  not 
change  the  number  of  variations  in  the  row  of  signs  (V,  1). 
Therefore,  when  x  passes  through  the  root  a,  Sturm's  functions 
lose  one  variation  of  signs,  and  only  one. 

In  the  same  way  it  may  be  shown  that  when  x  passes  through 
any  other  real  root  of  the  equation  f{x)  =  0,  Sturm's  functions 
lose  another  variation  of  signs. 

Cor.  1. — The  number  of  real  roots  of  the  equation  f(x)  =  0 
is  equal  to  the  number  of  variations  of  signs  lost  by  Sturm's  func- 
tions while  X  is  passing  from  —  oo  to  +  oo ;  the  number  of  real 
negative  roots  is  equal  to  the  number  of  variations  of  signs  lost 
while  X  is  passing  from  —  oo  to  0;  and  the  number  of  real  posi- 
tive roots  is  equal  to  the  number  of  variations  of  signs  lost  while 
X  is  passing  from  0  to  -f  oo . 


413 

CoK.  2. — Let  a  be  the  smallest  real  root  of  the  equation 
f{x)  =  0,  ^  the  next  greater,  c  the  next,  and  so  on. 

Just  after  x  passes  through  the  root  a,  f{x)  and  f'(x)  have 
like  signs;  and  just  before  x  passes  through  the  root  h,  f{x)  and 
f'{x)  have  contrary  signs  (VI).  But  f{x)  does  not  change  its 
sign  while  x  is  passing  from  a  to  h',  hence  f'{^)  must  change  its 
sign.  Therefore  the  equation  /'(a;)=0  has  one  real  root  between 
a  and  h.  In  the  same  way  it  may  be  shown  that  the  equation 
f'{x)  =z  0  has  one  real  root  between  b  and  c.  Therefore,  between 
any  two  consecutive  real  roots  of  the  equation  f(x)=:0  there  is 
one  real  root  of  the  equation  f'{x)  =  0. 

ScH. — The  sign  of  each  remainder  is  changed  in  order  that 
there  may  be  neither  a  gain  nor  a  loss  in  the  number  of  variations 
in  the  row  of  signs,  except  when  x  passes  through  a  real  root  of 
the  equation  (III-V). 

Find  the  number  and  situation  of  the  real  roots  of  the  follow- 
ing equations : 

1.    a^  —  3a^  —  ^x  -^13  =  0. 

f{x)=a^  —  3x^  —  4:X  +  13, 
fix)  =  3x^  —  6X—4.    (603), 

f^(x)  =  2x-6     (613), 

Mx)=-{-l. 

AS8T7HED  VALTTES  OP  X.  FUNCTIONS  AND  THEIR  SIGNS.  NtmBEB  OP  TAKIATIONS. 

fix),    fix),  f,ix),  f,ix). 


—  00 

— 

+ 

— 

+ 

3 

0 

+ 

— 

— 

+ 

2 

1 

+ 

— 

— 

+ 

2 

2 

+ 

— 

•— 

+ 

2 

3 

+ 

+ 

+ 

+ 

0 

+   00 

+ 

+ 

+ 

+ 

0 

Hence  all  the  roots  of  the  equation  are  real;  two  of  them  are 
positive  and  the  other  negative;  and  the  two  positive  roots  are 
situated  between  2  and  3. 

When  x  =  -'3   the  signs  of  the  functions  are    —   +    —   +? 
and  when  x=  —%  the  signs  of  the  functions  are    +    +   —    f ; 


414  THEORY    OF    EQUATIONS. 

hence  the  negative  root  is  between  —  2  and  —  3.  To  separate 
the  two  roots  which  he  between  2  and  3  we  must  substitute  for  x 
some  number  or  numbers  lying  between  2  and  3.  AVhen  x  =  2^ 
the  signs  of  the  functions  are  —  —  d:  +•  Here  we  have  only 
one  variation  whether  we  consider  the  vanishing  function  f^  (x) 
to  be  positive  or  negative;  hence  one  of  the  positive  roots  lies 
between  2  and  2J,  and  the  other  between  2^  and  3. 
2.    2^3  _  3.^  _  12.^^  _f_  24  =  0. 

Ans.  Three ;  one  between  1  and  2,  one  between  4  and  5, 
and  one  between    —  3   and   —  4. 


3. 

a:3  ^  6r^  +  10.T  —  1  =  0. 

4. 

x^  —  Qx^  -{- Sx  -\-  ^0  =  0. 

5. 

a:^  +  4  =  0. 

6. 

a^  —  2x^-{-3a^  —  W-\-Sx  —  d  =  0. 

7. 

x-i  —  92.-«  +  6a:*  +  15a^^  _  12ar5  -  7a;  +  6 

8. 

cc4  _|_  a;3  _  a4j  _  2x-\-  4  =  0. 

HORNER'S    METHOD    OF    APPROXIMATION. 
615.  Let  it  be  required  to  find  a  root  of  the  equation 
a:»  +  Aa:«-i  +  Bx"-2-f 4-Ka;  +  L  =  0    .    .     .     (1). 

Suppose  a  to  be  the  integral  part  of  the  root  required,  and 
r,  s,  t, . . . ,  ,  taken  in  order,  to  be  the  digits  of  the  fractional  part. 

Let  a  be  found  by  trial  (608)  or  by  Sturm's  Theorem ;  then 
find  an  equation  whose  roots  shall  be  less  by  a  than  those  of  (1) 
(598). 

Let  y«  +  Ay-i  +  By-2  +  . . .  -f  K>  +  U  =  0  . . .  (2)  be  that 
equation. 

In  this  equation  the  value  of  y  is  less  than  1 ;  hence  the  terms 
containing  the  higher  powers  of  y  are  comparatively  small;  neg- 
lecting these,  we  have,  approximately, 

'K'y  -f  L'  =  0,    whence    y  ^  —=-,, 

The  first  figure  in  the  value  of  y  is  r. 

Xow  find  an  equation  whose  roots  shall  be  less  by  r  than  those 
of  (2).  Let  ^«+A";z»-i+  B"z^-^+  ....  +K";2+L"=0.  .  .(3) 
be  that  equation. 


HORKER'S    METHOD    OF    APPROXIMATION.  415 

In  tliis  equation  the  value  of  z  is  less  than  .1 ;  hence,  we  have, 

L" 

approximately,  K";2  +  L"  =  0;  whence  2;  =  —  =7-,.  This  pro- 
cess may  be  continued  to  any  desired  extent,  and  we  shall  have 
finally  x^a-\-r-^s-\-t-\-,,.. 


RULE. 

I.  Find  the  integral  part  of  the  root  hy  Sturm's  Theorem  or 
otherwise. 

II.  Find  an  equation  whose  roots  shall  he  less  than  those  of  the 
given  equation  hy  the  integral  part  of  the  required  root 

III.  Divide  the  independent  term  of  the  transformed  equation 
hy  the  coefficient  of  the  adjacent  term^  change  the  sign  of  the  quo- 
tient and  write  the  first  figure  of  the  result  as  the  first  figure  of 
the  fractional  part  of  the  root, 

IV.  Find  an  equation  whose  roots  shall  he  less  than  those  of  the 
second  equation  by  the  first  figure  in  the  fractional  part  of  the 
required  root, 

V.  Divide  the  independent  term  of  this  transformed  equation 
hj  the  coefficient  of  the  adjacent  term,  change  the  sign  of  the  quo- 
tient, and  write  the  first  figure  of  the  result  as  the  second  figure  of 
the  fractional  part  of  the  required  root. 

VI.  Continue  this  process  until  tlw  root  is  obtained  to  the 
required  degree  of  accuracy. 

ScH.  1. — To  obtain  the  negative  roots  it  is  best  to  change  the 
signs  of  the  alternate  terms  of  the  given  equation,  and  then  find 
the  positive  roots  of  the  result;  changing  the  signs  of  these,  we 
obtain  the  negative  roots  required. 

ScH.  2. — If  a  trial  figure  of  the  root,  obtained  by  any  division, 
causes  the  two  last  terms  of  the  succeeding  equation  to  have  the 
same  sign,  that  figure  is  not  the  correct  one  and  must  be  changed. 

ScH.  3. — K  K'  should  reduce  to  zero  in  the  operation,  then 

/    T"' 
we  should  have,  approximately,  J'^2  +  L'=0 ;  whence  yz=^y  —^* 


416  THEORY    OF    EQUATIONS. 


JEXAJirPLES, 


1.  Find  one  root  of  the  equation   a:®  —  2a^  —  20.?-  —  40  =  0. 

By  Sturm's  Theorem  we  find  that  this  equation  has  only  one 
real  root,  and  that  the  integral  part  of  this  root  is  6.  We  now 
find  two  figures  of  the  fractional  part  as  follows : 


1-2 

—  20 

—  40   6.23 

+    6 

+  24 

+  24 

+    4 

+    4 

—  16^i> 

-f    6 

+  60 

+  10 

+  64<i> 

+    6 

+  16^*> 

1(1)  _,_  16(1) 

+  64<i> 

-  16<i> 

+    0.2 

+    3.24 

+  13.448 

+  16.2 

+  67.24 

—    2.552^*> 

+    0.2 

+    3.28 

+  16.4 

4-  70.52^«> 

+    0.2 

+  16.2^2] 

1<8>  +  16.6^2>  +  70.52^2>  —    2.552^«> 

We  find  the  coeflBcients  of  an  equation  whose  roots  are  less  hy 
6  than  those  of  the  given  equation,  using  the  method  explained 
in  Art.  598.  These  coeflBcients  are  1,  16,  64,  and  —  16,  marked 
(1)  in  the  operation.  Dividing  16  by  64,  we  obtain  .2,  which  is 
the  second  figure  of  the  root.  We  next  find  the  coefficients  of  an 
equation  whose  roots  are  less  by  .2  than  those  of  the  second  equa- 
tion. These  coeflBcients  are  marked  (2)  in  the  operation.  Divid- 
ing 2.552  by  70.52,  we  obtain  .03,  which  is  the  third  figure  of  the 
root!  This  process  may  be  continued  until  the  root  is  obtained 
to  any  required  degree  of  accuracy. 

2.  Find  one  root  of  the  equation  x^-{-a^—d0x^~20x—20=0. 

By  Sturm's  Theorem,  we  find  the  integral  parts  of  the  two 
real  roots  to  be  5  and  —  5.  Changing  the  signs  of  the  alternate 
terms  of  the  equation,  we  find  the  fractional  part  of  the  negative 
root  as  follows : 


HOENER'S    METHOD    OF    APPROXIMATION-. 


417 


1  —  1 

+  5 

—  30 
+  20 

+  20 
—50 

-  20  1  5.73 

—  150 

+  4 
+  5 

-  10 

+  45 

—30 

+  175 

— 170^i> 

+  9 

+  5 

+  35 
4-  70 

+  145^i> 

+  14 
+  5 


+  105^i> 


19(i> 


l(i)_^19(i) 
+   0.7 


+  105^i> 
+   13.79 


+  145(i> 
+  83.153 


— 170^i> 
+  159.7071 


+  19.7 
+     .7 


+  118.79 
+   14.28 


+  20.4 

+     .7 


+  133.07 

+  14.77 


+  228.153 
+  93.149 
+  321.302^2) 


10.2929(»> 


+  21.1 

+     .7 
21.8^2) 


+  147.84^«> 


1(«>  +  21.8<2>  +147.84(2>     +321.302^2)       _  10.2929(8> 
Hence  the  negative  root  of  the  given  equation  is  —  5.73  +. 
Find  the  real  roots  of  the  following  equations : 


3. 

a^^2Ti—23x—70c=0, 

Ans.   5.1345. 

4. 

a:3_ic2  4-70a:-300=0. 

Ans.   3.7387. 

5. 

a^4_a;2_500=0. 

Ans.  7.6172. 
(      3.3792, 

6. 

a^_a;2_40a;  + 108=0. 

Ans.   I      4.5875, 

—6.9667. 

(   1.7191, 

7. 

a:3_4c2_24a;+48=0. 

Ans.   \      6.5461, 
(  -4.2652. 

8. 

.^_l_a;3_|_^_a;_500=0. 

.    j   4.4604, 
^^^-  \  -4.9296: 

9. 

24_9^_lla^_20a;  +  4=0. 
27 

Ans     \        '^^^^^ 
^'''-  ]  10.2586. 

418 


THEORY    OF    EQUATION'S. 


Definitions 


CQ 

.  o 

X  < 

XI  P 

a 

EH 


OENERAIi  PbOPERTIES.    ^ 


616.  SYNOPSIS    FOR   REVIEW. 

The  general  equation  oftJiG  v}^-  degree. 
The  absolute  or  independent  term. 
A  function  of  a  quantity. 
A  rational  integral  function  of  x. 
<  A  root  of  the  equation  f  {x)  =^  0. 
When  f(x)  is  divisible  by  X  —  r. 
When  Visa  root  off(x)  =  0. 
Number  of  roots  of  f  {x)  =  0. 
Tofiiid  an  equation  whose  roots  are  given. 
Bdatim  between  the  coefficients  of  i\. 
fix)  and  the  roots  of  f{x)  =0.     j  3. 
Cor.  i,  2,  3,  4,  5,  6,  '  3. 

When  f{x)  =0   cannot  have  a  root 

which  is  a  rational  fraction. 

Boots  of  the  form  of  a  -^bV—l  and 

a  —  b  V—  1.     Cor.  i,  2,  3, 4. 
To  change  the  signs  of  the  roots  of  an 

equation. 
To   transform   an  equation  containing 

fractional  coefficients  into  another  in 

irhich  the  coefficients  are  integers,  that 

of  the  first  term  being  unity. 
To  transf/rm  an  equation  into  another, 

the  roots  of  which  differ  from  those  of 

the  given  equation  byagicen  quantity. 

Rule. 
To  cause  the  second  or  third  term  of  an 

equation  to  disappear.    Cor. 
Cor.  1,  2,  3. 
Primitive  function. 

First  deHvative,  Second  derivative,  etc. 
First  derivative  of  product  of  functions. 

Roots  common  to  two  equations.    Equal  Roots. 

r  Number  of  roots  of  f{x)  =  0  lying  be- 
Limits  of  the  Roots   I        tween  p  and  q.     Cor.  1,  2^  3. 
OF  AN  Equation.       j  Limits  of  positive  roots. 
I  Limits  of  negative  roots, 

Sturm's  Theorem.  .  .  3  ^'  ^^'  ^^^'  ^^'  ^'  ^^' 

\  Sch. 

Hornek's  Method  of  Approximation.    Mule. 


Transformation 
Equations. 


of 


Theorem  op  Descartes. 


Derived  Functions 


WEBSTER'S 

DICTIONARIES. 


UNABRIDGED  QUARTO,  NEW  EDITION.  1928  Pages,  8000  En- 
gravings. Over  4600  New  Words  and  Meanings,  Biographical 
Dictionary  of  over  9700  Names. 

NATIONAL   PICTORIAL,   OCTAVO.    1040  Pages,  600  Illustrations. 

COUNTING-HOUSE   DICTIONARY.     With  Illustrations. 

ACADEMIC   QUARTO.     834  I  llustrations. 

HIGH    SCHOOL   DICTIONARY.     297  Illustrations. 

COMMON    SCHOOL    DICTIONARY.     274  Illustrations. 

PRIMARY   SCHOOL   DICTIONARY.    204  Illustrations. 

POCKET  DICTIONARY.     With  Illustrations. 


WEBSTER  IS  THE  STANDARD  FOR  THE  ENGLISH 
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A  CA  DEM  Y,    IVES  T  FOIN  T. 

"\1T  A  RM  L  Y    Recommended    by      a  T  HAVE  looked,  so  that  I  might 
VV      Bancroft,  Prescott,  Motley,  1 


Geo.  p.  Marsh,  Halleck,  Whittier, 
A\ii,Lis,  Saxe,  Elihu  Burritt,  Daniel 
Webster,  Rufus  C:  oate,  H.  Cole- 
ridge, Smart,  Horace  Mann,  Presi- 
dents WooLSEV,  Wayland,  Hopkins, 
NoTT,  Walker,  Anderson  [mure  than 
FIFTY  College  Presidents  in  all], 
and  the  best  American  and  European 
Scholars. 

'HE  best  practical  English  Dic- 


aTHEi 

1       tic 


n(it  grow  wrong,  at  Webster's 
Dictionary,  a  work  ot  the  jgreatest 
learning,  researc'i  and  abiity."— Lord 
Chief  Justice  of  England,  in  1863. 
a  fpHE  Courts  look  to  it  as  of  the 
1  highest  authority  in  a'l  ques- 
tions of  definition." — Chief  Justice 
Waite,  U.  S.  Supreme  Court. 

(.(,  TT  has  received  the  highest  com- 
1  mendations  in  the  Courts  of 
England,  and  its  definitions  have  been 
universally  followed  in  the  Courts  of 
this  cowaXXY.^''— Albany  Law  yournal^ 
Quarterly  Review^  October^  1873.  J'^^y  1°,  1875. 

Indorsed  by  State  Superintendents  of  Schools  in  35  States. 

More  than  32.000  copies  of  Webster's  Unabridged  have  been  placed  in 
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as  far  as  we  are  aware,  lia«  ever  piibllciy  reco<rnized.  any  o'lier 
Dictionary  tlian  \Veb*«fer  as  its  standard  of  ortlioj^raplty, 
with  the  s1nsl<^  exception  of  the  publishers  of  another  Dictionary.— While 
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^HE  ATTENTION  OF  TEACHERS,  SCHOOL 

Officers,  and  others,  is  called  to  the  superior  adaptability  of  the  Spen- 
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Durability^  Elasticity^  and  Evenness  of  Point. 


They  are  made  by  the  best  workmen  in  Europe,  and  of  the  best  materials, 
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WELL-KNOWN  Mo.    I,    VlOVe  t/ia7l   8,000,000  are  annually  sold. 

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which,  in  connection  with  our  SPENCERIAN  brand,  gives  us  the  control  of 
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As  Familiar  to  the  Schools  of  the   United  States  as 
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Robinson's  Progressive  Course 


OF 


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ROBINSON'S  PROGRESSIVE  COURSE  OF 
MATHEMATICS,  being  the  most  complete  and  scientific  course  of 
Mathematical  Text-books  published,  is  more  extensively  used  in  the  Schools 
and  Educational  Institutions  of  the  United  States  than  any  compeLing  series. 

In  its  preparation  two  objects  were  kept  constantly  in  view  :  First,  to  fur- 
nish a  full  ana  complete  Series  of  Text-Books,  which  should  be  sufBcientto  give 
the  pupil  a  thorough  and  practical  business  education  ;  Second^  to  secure  that 
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Robinson  s  Shorter  Course. 

In  order  to  meet  a  demand  from  many  quarters  for  a  Series  of  Arithmetics, 
few  in  number  and  comprehensive  i  n  character,  we  have  published  the  above 
course,  in  TIVO  books,  in  which  Oral  and  Written  Arithmetic  is  combined. 
These  books  have  met  with  very  great  popularity,  having  been  introduced  into 
several  of  the  largest  cities  in  the  United  States.  They  are  unusually  hand- 
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past,  the  several  series  of  Readers  submtte  !  for  their  examination,  they  have 
come  to  the  conclusion,  with  en  ire  unanimity,  tiiat  the  Educational  Series  p/ 
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seems  to  co.mbm  •  a  grrealer  number  of  merits  and  advantages  than  any  other 
scries  which  the  committee  h  xve  seen." 

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merely  placed  up^iT  a  list  to  be  used  at  discretion,  but  every  pupil  in  the  grades 
into  which  it  has  been  introduced,  has  and  uses  the  series." 


Catiicart's  Literary  Reader. 

TYPICAL    SELECTIONS 

from  the  Best  Authors ,  luith  Biographical  and  Critical  Sketches,  and 

numerous  notes.     Cloth.     438  pages. 

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Descriptive  circulars  and  price  lists  on  application.    The  most  liberal  terms 
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A   GOLD  MEDAL  was  atvardedio  Professor  SWINTON  at  the  Paris  Ex- 

J>osition  ty^jSyS,  as  an  author  c/  School  Text-j  ooks.  he  being;  the 

only  Atnerican  A  utiior  thus  higitly  iionored. 


Standard  Text-Books 

By 

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SWINTON'S  WORD-BOOK  SE^RIES.  The  only  perfectly 
graded  Series  of  Spellers  ev- r  made,  and  the  cheapest  in  the  market.  In  use 
in  more  than  10,000  Schools. 

Word  Primer.    A  Beginner's  Book  in  Oral  and  Written  Spelling:.  96pajjes, 
Word-Book  of  Spellins:.     Oral  and  Written.    Designed  to  attan  prac- 
tical results  i  i  the  acquisition  of  the  ordinary  English  vocabulary,  and  to 
serve  as  an  introduc.ion  to  Word  Analysis.     154  pages. 
Word  AualyslM.  A  Graded  Class-book  of  English  Derivative  Words,  with 

t radical  exercises  in  Spellinjf,  Analyzing,  Definmg,  Synonyms,  and  the 
Tse  of  Words,     i  vol.    128  pages. 
SWi:\TO>'S    HI«TOICI1i:S.    These  books  have  attained  great  pop- 
ularity.    A  new  edition  of  the  "Condensed"  has  just  been  issued,  i  a  wlich 
the  work  has  been  brought  djwn  to  the  present  time,  and  six  colored  maps  hare 
been  added. 

Primary  HiHtory  of  U.   S.    First  Lessons  in  our  Country's  History, 
bringing  out  the  salient  points,  and  aiming  to  combine  simplicity  with 
sense.    I  vol.  square,  fully  illustra' ed. 
Condensed  Seliool  History  of  U.  S.    A  Conden-ed  Schol  History 
of  the  United  State ;,  constructed  for  definite  results  in  Re^  itation,  and  con- 
taining a  new  method  of  Topical  Reviews.    New  edition,  brou'jh.  down  to 
the  present  time.     Illustrated  with   Maps,  many  of  wnich  are  colored, 
Portraits  and  Illustrations.    1  vol.  cloth.     300  pages. 
Outlines  of  the  World's  Ifl«tory.     Ancient,  Mediaeval  and  Modern, 
with  special  reference  to  the  History  of  Mankind.    A  most  excellen:  work 
for  the  proper  introduction  of  youth  into  the  study  of  General  History. 
1  vol  .  With  numerous  maps  and  illustrations.    500  pages,  lamo. 
S  '.yiNTON'S  GFIOGIl APHICAIi  COURSE.    The  famous  "two 
book  series,"  the  freshest,  best  graded,  most  beautilul  and  cheapest  Geographi- 
cal Course  ever  published.    Of  the  lar.e  cities  that  have  adopted  Swinton's 
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ton, S.  C,  Lancaster,  Pa.,  VN'illiamsport,  Pa.,  Macon,  Ga.   I  iround  numbeis, 
they  have  been  adopted  in  more  than  l.OOO  Cities  and  Towns  in  all  parts  of 
the  countrv,  and  h.ive,  with  inarkjd  P'e/irence,hcQn  made  the  bas  s  oi  Pro- 
fessional  Training  in  the  Leading  Normal  Schools  of  the  United  States. 
Elementary   Course  in   Oeojjraphy.    Desifjned  as  a  class-book  for 
primary  and  intermediate  grades  ;  ancf  as  a  complete  Shorter  Course  for  un- 
graded schools.     123  pages,  8vo. 
Complete  Course  In  (ieosrapliy.    Physical,  Industrial  and  Political; 
with  a  special  Geography  for  each  State  in  the  Union.    Designed  as  a  class- 
book  far  intermediate  and  grammar  grades.     136  pages,  4to. 

The  Maps  in  both  books  possess  novel  features  of  tne  highest  practical 
value  in  education. 

S\ri\TON'S  RAHIBIiES  AI?IONG  WORDS:  Their  Poetry, 
History  and  Wisdom.  A  Standard  Work  to  all  who  love  tne  riches  of  the 
English  Language.  By  William  Swin ton,  M.A.  Hands  mely  bound  in  flexible 
cloth  and  marbled  edges.    302  pages. 

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thus  obtainable,  we  will  supply  them,  transportation  paid,  at  liberal  rates.  De- 
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Very  liberal  terms  for  introduction^  exchange  and  examination. 

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NEW  YORK  and  CHICAGO. 


Approved  Text-Books 

FOR 

HIQH   SCHOOLS. 


THERE  are  no  text-books  that  require  in  tlieir  preparation  so  much  prac- 
tical scliolarship,  combined  with  the  teacher's  experience,  as  those  com- 
piled for  use  in  High  Schools,  Seminaries  and  Colleges.  The  treatment  must 
be  succinct  yet  thorough  ;  accuracy  of  statement,  clearness  of  expression,  and 
scientific  grad;ition  are  indispensable.  We  have  no  special  claims  to  make  for 
our  list  on  the  score  of  the  Ancient  Classics,  but  in  the  modern  languages, 
French,  German  and  Spanish,  in  Botany,  Geology,  Chemistry  and  Astronomy, 
and  the  Higher  Mathematics,  Moral  and  Mental  Science,  etc.,  etc.,  we  chal- 
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scholars,  and  are  reprinted  abroad,  while  others  have  enjoyed  a  National  repu- 
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WOODBURY'S  GERMAN  COURSE.  Comprising  a  full  series,  from 
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FASQUEf.LE'S  FRENCH  COURSE.  On  the  plan  of  Woodbury's 
Method  ;  also  a  complete  series. 

MIXER'S  MANUAL  OF  FRENCH  POETRY. 

LANGUELLIER  db  MONSANTO' S  FRENCH  GRAMMAR. 

UENNEQUIN'S  FRENCH   VERBS. 

MONSANTO' S  FRENCH  STUDENT'S  ASSIS7ANT. 

MONSANTO  it  LANOUELLIER'S  SPANISH  GRAMMAR. 

GRA  Y'S  BOTANICAL  SERIES 

DANA'S  WORKS  ON  GEOLOGY. 

ELIOT  db  STORER'S  CHEMISTRY. 

ROBINSON'S  HIGHER  MATHEV-i  TICS. 

SWINTON'S  OUTLINES  OF  HISTORY. 

WILLSON'S  OUTLINES  OF  HISTORY. 

CATHCARTS  LITERARY  READER. 

HICKOK'S  WORKS  ON  METAPHYSICS. 

HUNTS  LITERATURE  OF  THE  ENGLISH  LANGUAGE. 

WELLS'    WORKS  ON  NATURAL  SCIENCE. 

KERVS  COMPREHENSIVE  ENGLISH  GRAMMAR. 

WEBSTER'S  ACADEMIC  DICTIONARY. 

TOWNSEND'S  CIVIL  GOVERNMENT. 

WHITE'S  DRA  WING. 

TAYLOR-K'UHNER'S  GREEK  GRAMMAR. 

KIDDLE'S  ASTRONOMY. 

SWINTON'S  COMPLETE  GEOGRAPHY.  Etc.  E*c. 


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